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$ax^2 + bx + c = 0$

## 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.

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

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.

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

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.

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

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

Cite this content, page or calculator as:

Furey, Edward "Quadratic Formula Calculator"; CalculatorSoup, https://www.calculatorsoup.com - Online Calculators