10-4 Solving Quadratic Equations by Using the Quadratic Formula
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 />
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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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