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Quadratic Formula Calculator
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Example: 2x^2-5x-3=0
Step-By-Step Example
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Example (Click to try)
2x2−5x−3=0
About the quadratic formula
Solve an equation of the form ax2+bx+c=0 by using the quadratic formula:
x=
−b±√b2−4ac
2a
Quadratic Formula Video Lesson
Solve with the Quadratic Formula Step-by-Step [1:29]
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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
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Step-by-Step Examples
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Find the Roots (Zeros)
Step 1
Set equal to .
Step 2
Solve for .
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Set the equal to .
Solve for .
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Add to both sides of the equation.
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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