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10-4 Solving Quadratic Equations by Using the Quadratic Formula

10-4 Solving Quadratic Equations by Using the Quadratic Formula

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<strong>10</strong>-4 <strong>Solving</strong> <strong>Quadratic</strong><br />

<strong>Equations</strong> <strong>by</strong> <strong>Using</strong> <strong>the</strong><br />

<strong>Quadratic</strong> <strong>Formula</strong><br />

This presentation was created following <strong>the</strong> Fair Use<br />

Guidelines for Educational Multimedia. Certain materials are<br />

included under <strong>the</strong> Fair Use exemption of <strong>the</strong> U. S. Copyright<br />

Law. Fur<strong>the</strong>r use of <strong>the</strong>se materials and this presentation is<br />

restricted.


Objectives<br />

• Students will solve quadratic equations<br />

<strong>by</strong> using <strong>the</strong> <strong>Quadratic</strong> <strong>Formula</strong>.<br />

• Students will use <strong>the</strong> discriminant to<br />

determine <strong>the</strong> number of solutions for a<br />

quadratic equation.<br />

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Vocabulary<br />

• quadratic formula:<br />

x<br />

=<br />

−<br />

b<br />

±<br />

b<br />

2 − 4ac<br />

2a<br />

• discriminant: <strong>the</strong> expression under <strong>the</strong><br />

radical sign(b 2 – 4ac). If <strong>the</strong> discriminant is:<br />

• negative: no real roots<br />

• positive: 2 real roots<br />

• zero: one real root<br />

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Use two methods to solve<br />

Method 1<br />

Factoring<br />

Original equation<br />

Factor<br />

or<br />

Zero Product Property<br />

Solve for x.<br />

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Method 2<br />

For this equation,<br />

<strong>Quadratic</strong> <strong>Formula</strong><br />

<strong>Quadratic</strong> <strong>Formula</strong><br />

Multiply.<br />

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Add.<br />

Simplify.<br />

or<br />

Answer: The solution set is {–5, 7}.<br />

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Use two methods to solve<br />

Answer: {–6, 5}<br />

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Solve<br />

<strong>by</strong> using <strong>the</strong> <strong>Quadratic</strong> <strong>Formula</strong>.<br />

Round to <strong>the</strong> nearest tenth if necessary.<br />

Step 1<br />

Rewrite <strong>the</strong> equation in standard form.<br />

Original equation<br />

Subtract 4 from<br />

each side.<br />

Simplify.<br />

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Step 2<br />

Apply <strong>the</strong> <strong>Quadratic</strong> <strong>Formula</strong>.<br />

<strong>Quadratic</strong> <strong>Formula</strong><br />

and<br />

Multiply.<br />

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Add.<br />

or<br />

3/11/2008 <strong>10</strong>


Solve<br />

<strong>by</strong> using <strong>the</strong> <strong>Quadratic</strong> <strong>Formula</strong>.<br />

Round to <strong>the</strong> nearest tenth if necessary.<br />

Answer: {–0.5, 0.7}<br />

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State <strong>the</strong> value of <strong>the</strong> discriminant for .<br />

Then determine <strong>the</strong> number of real roots of <strong>the</strong><br />

equation.<br />

and<br />

Simplify.<br />

Answer: The discriminant is –220. Since <strong>the</strong> discriminant<br />

is negative, <strong>the</strong> equation has no real roots.<br />

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State <strong>the</strong> value of <strong>the</strong> discriminant for .<br />

Then determine <strong>the</strong> number of real roots of <strong>the</strong><br />

equation.<br />

Step 1<br />

Rewrite <strong>the</strong> equation in standard form.<br />

Original equation<br />

Add 144 to<br />

each side.<br />

Simplify.<br />

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Step 2<br />

Find <strong>the</strong> discriminant.<br />

and<br />

Simplify.<br />

Answer: The discriminant is 0. Since <strong>the</strong> discriminant is 0,<br />

<strong>the</strong> equation has one real root.<br />

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State <strong>the</strong> value of <strong>the</strong> discriminant for .<br />

Then determine <strong>the</strong> number of real roots of <strong>the</strong><br />

equation.<br />

Step 1<br />

Rewrite <strong>the</strong> equation in standard form.<br />

Original equation<br />

Subtract 12 from<br />

each side.<br />

Simplify.<br />

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Step 2<br />

Find <strong>the</strong> discriminant.<br />

and<br />

Simplify.<br />

Answer: The discriminant is 244. Since <strong>the</strong> discriminant<br />

is positive, <strong>the</strong> equation has two real roots.<br />

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State <strong>the</strong> value of <strong>the</strong> discriminant for each<br />

equation. Then determine <strong>the</strong> number of real<br />

roots for <strong>the</strong> equation.<br />

a.<br />

Answer: –4; no real roots<br />

b.<br />

Answer: 0; 1 real root<br />

c.<br />

Answer: 120; 2 real root<br />

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