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A quadratic equation is a polynomial equation of degree two,
which can be written in the form ax2 + bx + c = 0, where x is a variable and a, b and c are constants with a ≠ 0. |
 Solving quadratic equations:
Solving a quadratic equation can range from being a simple task, to being a challenge. These refresher pages build an arsenal of strategies for solving quadratic equations. |
Quadratic Equation Strategies:
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To solve a quadratic equation by factoring:
For more details on the process of factoring, see the Algebra 1 Section on Factoring.
The information presented on this page is meant as a refresher of those factoring skills, as they pertain to quadratic equations.
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Factoring Method |
1. |
Express the equation in the form ax2 + bx + c = 0. |
2. |
Factor the left hand side (if 0 is on the right). |
3. |
Set each of the two factors equal to zero. |
4. |
Solve for x to determine the roots (or zeros). |
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Simple quadratic equations with rational roots can be solved by factoring. Let's refresh our memories on factoring these simple quadratic equations as they appear in different situations.
See Algebra 1 Factoring for more examples.
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Examples of Solving Quadratic Equations by Factoring:
Factoring with GCF
(greatest common factor):
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Solve: 4x2 - 28x = 0
4x(x - 7) = 0
4x = 0; x - 7 = 0
x = 0; x = 7
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Find the largest value which can be factored from each term on the left side of the quadratic equation.
The roots (zeros) correspond to the locations of the x-intercepts of the function y = 4x2 - 28x. |
Factoring Trinomial with Leading Coefficient of One:
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Solve: x2 + 2x - 15 = 0
(x + 5)(x - 3) = 0
x + 5 = 0; x - 3 = 0
x = -5; x = 3
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When the leading coefficient is one, the product of the roots will be the constant term, and the sum of the roots will be the coefficient of the middle x-term. |
Factoring Difference of Two Squares:
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Solve: x2 - 81 = 0
(x + 9)(x - 9) = 0
x + 9 = 0; x - 9 = 0
x = -9; x = 9
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Remember the pattern for the difference of two squares, where the factors are identical except for the sign between the terms.
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Factoring Trinomial with Leading Coefficient Not One:
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Solve: 3x2 + 11x - 4 = 0
(3x - 1)(x + 4) = 0
3x - 1 = 0; x + 4 = 0
x = 1/3; x = -4
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Life gets more difficult when the leading coefficient is not one. Check out the Factoring section to see more strategies for factoring these pesky problems. |
Where's the x2 ?
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Solve: 2x(x + 4) = x - 3
2x2 + 8x = x - 3
2x2 + 7x + 3 = 0
(2x + 1)(x + 3) = 0
2x + 1 = 0; x + 3 = 0
x = - 1/2; x = -3
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Sometimes you have to "work" on the equation to get the needed quadratic form. In this case, distribute, and the x2 will appear. |
Dealing with Proportions:
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Solve: 
2( x + 4) = ( x + 1)( x - 2)
2 x + 8 = x2 - x - 2
0 = x2 - 3 x - 10
0 = ( x - 5)( x + 2)
x - 5 = 0; x + 2 = 0
x = 5; x = -2
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x2 may appear when cross multiplying ("product of the means equals product of the extremes") is employed in a proportion. |
To solve a quadratic equation by taking a square root:
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Square
Root Method (ax2 is the only variable term) |
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Isolate the ax2 term on one side of the equation. |
2. |
Take the square root of both sides. |
3. |
Remember to use ±, as there are two solutions. |
4. |
Express the roots (or zeros). |
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The square root method applies only to one specific situation. This method works when there is no middle bx-term in the equation. The only variable in the equation is an x2-term.
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Examples of Solving Quadratic Equations by Square Root Method:
Easiest Set-Up (has "x2 = __"): |
Solve: x2 = 121
x = ±11
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We saw this example under the previous method of factoring. Here it is again, being solved by the square root method. Notice the answers are the same.
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Difference of Two Squares:
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Solve: x2 - 81 = 0
x2 = 81
x = ± 9
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We saw this example under the previous method of factoring. Here it is again, being solved by the square root method. Notice the answers are the same.
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Only variable is the x2 term, with subtraction of constant. |
Solve: x2 - 12 = 0
x2 = 12
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This example shows the numeric value (the constant), subtracted from the x2 term.
Approximate decimal answers:
x = -3.464102 and
x = 3.4641016 |
Only variable is the x2 term, with addition of constant. |
Solve: x2 + 9 = 0
x2 = -9
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This example shows the numeric value (the constant), added to the x2 term, which creates "complex" answers containing the imaginary "i". |
There are tools more powerful than factoring and square root method
for solving quadratic equations.
Find refreshers on these other strategies for solving quadratic equations:
completing the square and quadratic formula. |
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For calculator help with graphing parabolas
click here. |
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