Quadratic Formula Calculator

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\[ ax^2 + bx + c = 0 \]

a =

b =

c =

Answer:

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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