Write quadratic equation 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

What are quadratic functions?

Quadratic functions are functions of the form . This means, there is no x to a higher power than . The graph of a quadratic function is a parabola.

How do you write a quadratic equation in standard form?

How do you write quadratic equations in standard form give 3 examples?