Express quadratic function in standard form calculator

The calculator below solves the quadratic equation of

ax2 + bx + c = 0

.

In algebra, a quadratic equation is any polynomial equation of the second degree with the following form:

ax2 + bx + c = 0

where x is an unknown, a is referred to as the quadratic coefficient, b the linear coefficient, and c the constant. The numerals a, b, and c are coefficients of the equation, and they represent known numbers. For example, a cannot be 0, or the equation would be linear rather than quadratic. A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing. Only the use of the quadratic formula, as well as the basics of completing the square, will be discussed here (since the derivation of the formula involves completing the square). Below is the quadratic formula, as well as its derivation.

Derivation of the Quadratic Formula

From this point, it is possible to complete the square using the relationship that:

x2 + bx + c = (x - h)2 + k

Continuing the derivation using this relationship:

Recall that the ± exists as a function of computing a square root, making both positive and negative roots solutions of the quadratic equation. The x values found through the quadratic formula are roots of the quadratic equation that represent the x values where any parabola crosses the x-axis. Furthermore, the quadratic formula also provides the axis of symmetry of the parabola. This is demonstrated by the graph provided below. Note that the quadratic formula actually has many real-world applications, such as calculating areas, projectile trajectories, and speed, among others.

Calculator Use

This online calculator is a quadratic equation solver that will solve a second-order polynomial equation such as ax2 + bx + c = 0 for x, where a ≠ 0, using the quadratic formula.

The calculator solution will show work using the quadratic formula to solve the entered equation for real and complex roots. Calculator determines whether the discriminant \( (b^2 - 4ac) \) is less than, greater than or equal to 0.

 When \( b^2 - 4ac = 0 \) there is one real root.

 When \( b^2 - 4ac > 0 \) there are two real roots.

 When \( b^2 - 4ac < 0 \) there are two complex roots.

Quadratic Formula:

The quadratic formula

\( x = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a } \)

is used to solve quadratic equations where a ≠ 0 (polynomials with an order of 2)

\( ax^2 + bx + c = 0 \)

Examples using the quadratic formula

Example 1: Find the Solution for \( x^2 + -8x + 5 = 0 \), where a = 1, b = -8 and c = 5, using the Quadratic Formula.

\( x = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a } \)

\( x = \dfrac{ -(-8) \pm \sqrt{(-8)^2 - 4(1)(5)}}{ 2(1) } \)

\( x = \dfrac{ 8 \pm \sqrt{64 - 20}}{ 2 } \)

\( x = \dfrac{ 8 \pm \sqrt{44}}{ 2 } \)

The discriminant \( b^2 - 4ac > 0 \) so, there are two real roots.

Simplify the Radical:

\( x = \dfrac{ 8 \pm 2\sqrt{11}\, }{ 2 } \)

\( x = \dfrac{ 8 }{ 2 } \pm \dfrac{2\sqrt{11}\, }{ 2 } \)

Simplify fractions and/or signs:

\( x = 4 \pm \sqrt{11}\, \)

which becomes

\( x = 7.31662 \)

\( x = 0.683375 \)

Example 2: Find the Solution for \( 5x^2 + 20x + 32 = 0 \), where a = 5, b = 20 and c = 32, using the Quadratic Formula.

\( x = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a } \)

\( x = \dfrac{ -20 \pm \sqrt{20^2 - 4(5)(32)}}{ 2(5) } \)

\( x = \dfrac{ -20 \pm \sqrt{400 - 640}}{ 10 } \)

\( x = \dfrac{ -20 \pm \sqrt{-240}}{ 10 } \)

The discriminant \( b^2 - 4ac < 0 \) so, there are two complex roots.

Simplify the Radical:

\( x = \dfrac{ -20 \pm 4\sqrt{15}\, i}{ 10 } \)

\( x = \dfrac{ -20 }{ 10 } \pm \dfrac{4\sqrt{15}\, i}{ 10 } \)

Simplify fractions and/or signs:

\( x = -2 \pm \dfrac{ 2\sqrt{15}\, i}{ 5 } \)

which becomes

\( x = -2 + 1.54919 \, i \)

\( x = -2 - 1.54919 \, i \)

calculator updated to include full solution for real and complex roots

An online vertex form calculator helps you to find the vertex of a parabola and the vertex form of a quadratic equation. This vertex to standard form calculator quickly displays vertex and y-intercept points with a graph. Also, you can find how to find the vertex of a quadratic function, quadratic to vertex form, and vertex to standard form conversions in the context below.

Let’s start with some basics!

What is the Vertex Form?

In the conic section, the vertex form of a parabola is a point or place where it turns, it is also known as a turning point. If the quadratic function converts to vertex form, then the vertex is (h, k).

The vertex equation is

$$ y = a(x – h)^2 + k $$

What is the vertex of a parabola?

“The point at the intersection of the parabola and its line is a symmetry known as the vertex of the parabola”.

How to find the vertex of a parabola?

The vertex of a parabola is a specific point that represents the different values of the quadratic curve. The vertex can be either maximum (when parabola going downward) or minimum (when parabola going up). Therefore, the vertex form is the intersection of a parabola with its symmetric axis.

Normally, the vertex is (h, k), where h indicates the x-coordinates, and k stands for y-coordinates.

A standard form of a parabola \( ax^2 + bx + c \), so we can use quadratic equations of the vertex coordinates:

$$ h = -b / 2a $$

$$ k = c – b^2 / 4a $$

The above equations use our online standard to vertex form calculator to make calculations precisely.

However, an Online Parabola Calculator helps to find the standard form and vertex form of a parabola equation for the given values.

Example:

Finding the vertex of a parabola for the equation:

$$ = 2(x – (-6))^2 – 13 $$

Solution:

According to given equation

Vertex form is:

$$ y = 2 (x + 6)^2 – 13 $$

Standard form of given equation is:

$$ y = 2 x^2 + 24 x + 59 $$

Where,

Characteristic Points are:

Vertex P (-6, -13)

Y-intercept P (0, 59)

An online parabola vertex calculator can display a parabola graph with exact values when you substitute the same values for a vertex form equation.

How to Convert Standard form to Vertex form:

The standard form of a quadratic equation is \( m = a x^2 + b x + c \), where m and x are variables and a, b, and c are the coefficients. It is simple to solve an equation when it is in standard form because we calculate the answer with a, b, and c. However, when you need a graph of a parabola, quadratic function. The process is smooth when the equation is in vertex form. The standard to vertex form of a quadratic equation is \( Q = m(x – h)^2 + K \), where m represents the slope. Our standard form to vertex form calculator can change the standard to vertex form. Now get ready to know how to find vertex from standard form. Well, if you want to do it manually then follow these instructions:

• Write the standard form of a quadratic function: \( m = a x^2 + b x + c \).
• Divide first two terms by a: \( m = a (x^2 + b/a x) + c \).
• Complete the square for the expression with x. Then, add and subtract \( (b/(2a))^2 \) from the equation: \( m = a [ x^2 + x (b/a) + (b/(2a))^2 – (b/(2a))^2] + c\).
• Now, according to square formula, we can say write: \( m = a [(x + (b/(2a))^2 – (b/(2a))^2] + c \) and multiply the terms with a: \( m = a (x + (b/(2a))^2 – b^2/4a + c \).
• Then compare the vertex equation: \( m = a (x – h)^2 + K \) , the vertex of parabola is: \( h = – b / 2a and k = c – b^2 / 4a \).

However, an Online Slope Calculator helps to find the slope (m) or gradient between two points in the Cartesian coordinate plane.

How to Convert vertex form to standard form:

A free online vertex form calculator can convert vertex form to the standard form of a parabola. If you want to know how to change the vertex to standard form, let’s start!

• Write an equation in vertex form: \( m = a (x – h)^2 + K. \)
• Now, expand the square formula: \( m = a (x^2 + y^2 + 2hx) + K. \)
• Multiply the inner side or bracket: \( a x^2 + a y^2 + 2 ahx + K. \)
• Then, compare with quadratics in vertex form of a parabola: \( m = a x^2 + b x + c. \)
• Estimate the values of parameter: \( b = – 2 ah and c = a h^2 + k \).

How Vertex Form Calculator Works?

This vertex calculator can convert to vertex form or standard form with these steps:

Input:

• First, select standard to vertex form or vertex form to standard form from the drop-down list.
• Now, the vertex form of a parabola calculator displays an equation according to the selected option.
• Then, substitute the value of variables according to the equation.
• Click the calculate button to see the conversion and vertex points.

Output:

• This vertex of a parabola calculator displays a vertex and standard form of the given equation.
• This parabola to vertex form calculator also provides characteristic points with a parabola graph.

FAQ:

What is the vertex of an angle?

The vertex of an angle is the endpoint of two different rays that form the angle.

What is the common vertex?

A common vertex is shared by two angles. A vertex is a point of intersection where two linear construction lines intersect each other. However, the vertex form of a quadratic function calculator finds the common vertex of the parabola.

How we can find the turning point of a function?

A turning point of a line or function is a point where f′(x)=0. A turning point is a point where the parabola is upward (from decreasing to increasing) and f′(x)=0 at the point.

How we can determine the vertex with zeros?

First, find the zeros (0) by any factoring or the Quadratic Formula method. Now, find the x of the vertex by averaging the zeros. Then, we can calculate the f(x) to find out the y-coordinate of a vertex. Besides that, our online quadratic to vertex form calculator is the best way to calculate the vertex with zeros (0).

Final Thoughts:

Use this vertex form calculator to find the vertex and y-intercept points of the given equation. A special form of a quadratic function is a vertex form. With a parabola in vertex form calculator, we can see where the point of the parabola is maximum or minimum.

Reference:

From the source of Wikipedia: Etymology, Coefficients, Variables, The one-variable case, Bivariate case, Forms of a univariate quadratic function, Graph of the univariate function.

From the source of Virtual Nerd: Vertex, Maximum and minimum points, Roots of the univariate function, Exact roots, Upper bound on the magnitude of the roots.

From the source of Math Bits Notebook: Iteration, Bivariate (two-variable) quadratic function, Minimum/maximum, Exact roots.

What is the standard form of a quadratic equation calculator?