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Why is this concept useful?: The Quadratic equation is a popular type of equation that is seen in almost all areas of discipline whether it is Physics or Chemistry or Biology. The formula is useful because it gives us a way to find the solution to the equation in a matter of a few minutes without having out factor it out manually.
Where does this concept fit into the curriculum?: Algebra 1 and up

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How can we use the concept:

There are multiple ways to solve a quadratic equation but sometimes you don’t know the best method until you fail at the method you chose. The Quadratic Formula will always help you solve any quadratic equation. It works every time!

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

Sample Math Problems

1. The standard form of a quadratic equation is:

ax^{2} +bx+c=0

Use the Quadratic Formula to find the solutions for:

x^{2} +6x+9=0

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Practice Math Problems

1. Use the Quadratic Formula to find the solutions for:

x^{2} +x-12=0

2. Use the Quadratic Formula to find the solutions for:

\begin{array}{l} 4x^{2} +6x+2=0\\ \end{array}

3. Use the Quadratic Formula to find the solutions for

x^{2} +5x-6=0

4. Use the Quadratic Formula to find the solutions for:

2x^{2} +8x+2=0

5. What are the two mistakes in setting up the Quadratic Formula in the following problem?

Problem: Solve

2x^2 - x - 6 = 0

Set-up for the problem:

x = \frac{-1 \pm \sqrt{ (-1)^2 - 4(2)(6)}}{2(2)}

Solution:

Step 1: Determine the values for a, b and c; a = 1; b = 6, and c = 9.

Step 2:

x = \frac{-6\pm \sqrt{6^{2} -4( 1)( 9)}}{2( 1)}

= \frac{-6\pm \sqrt{36-36}}{2} =\frac{-6}{2} =-3

Therefore, there is one real root x = -3 or (-3, 0).

2. Use the Quadratic Formula to find the solutions for:

4x^{2} +5x-6=0

Solution:

Step 1: Determine the values for a, b and c; a = 4; b = 5 and c = -6.

Step 2: Plug into Quadratic Formula:

\frac{-5\pm \sqrt{5^{2} -4( 4)( -6)}}{2( 4)} =x=\frac{-5\pm \sqrt{25-( -96)}}{8} =\frac{-5+\sqrt{121}}{8}

Step 3: Now you have to split the main problem into two mini problems:

\frac{-5+11}{8} =\frac{6}{8} =\frac{3}{4}

\frac{-5-11}{8} =\frac{-16}{8} =-2

There are two roots for this equation, ¾ and -2. (¾ , 0) and (-2, 0).

3. Use the Quadratic Formula to find the solutions for:

3x^{2} -5x-2=0

Solution:

Step 1: Determine the values for a, b and c; a = 3; b =- 5 and c = -2.

Step 2: Plug into Quadratic Formula:

\frac{-5\pm \sqrt{5^{2} -4( 3)( -2)}}{2( 3)} =x=\frac{-5\pm \sqrt{25-( -24)}}{6} =\frac{-5+\sqrt{49}}{6}

Step 3: Now you have to split the main problem into two mini problems:

\frac{-5+7}{6} =\frac{2}{6} =\frac{1}{3}

\frac{-5-7}{6} =-\frac{12}{6} =-2

There are two roots for this equation, ⅓ and -2. (⅓ , 0) and (-2, 0).

4. Use the Quadratic Formula to find the solutions for:

x^{2} +6x-2=0

Solution:

Step 1: Determine the values for a, b and c; a = 1; b = 6 and c = -2.

Step 2: Plug into Quadratic Formula:

x = \frac{-6\pm \sqrt{6^{2} -4( 1)( -2)}}{2( 1)}

=\frac{-5\pm \sqrt{36-( -8)}}{2} =\frac{-5+\sqrt{44}}{2}

Step 3: Simplify the Radical since the square root of 44 is not perfect:

x=\frac{-6}{2} \pm \frac{2\sqrt{11}}{2}

Step 4: Simplify the Fractions:

-3\pm \sqrt{11}

Step 5: Split the problem:

-3+\sqrt{11} \approx 0.316

-3-\sqrt{11} \approx 6.316

The two roots for this equation are 0.316 and -6.316

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Quadratic Formula

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Sample Math Problems

Download FREE Math Resources

8 Math Hacks and Tricks to Turn Your 'Okay' Math Student Into a Math Champion!

How to Make Sure Your Child is Highly Successful and Becomes a Millionaire

Practice Math Problems