Chapter 11: Q. 99 (page 775)

If r1 and r2 are two solutions of a quadratic equation ax2 + bx + c = 0, it can be shown that

$r_{1} + r_{2} = - \frac{b}{a} a n d r_{1} r_{2} = \frac{c}{a}$
Solve this system of equations for r1 and r2.

Short Answer

Expert verified

$r_{1} = \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} r_{2} = {\frac{- b - \sqrt{b^{2} - 4 a c}}{2 a}}_{}$

Step by step solution

Step 1. Given information

r1 and r2 are two solutions of a quadratic equation ax2 + bx + c = 0

and $r_{1} + r_{2} = - \frac{b}{a} a n d r_{1} r_{2} = \frac{c}{a}$

Step 2. r1 in terms of r2

First we divide the equation $r_{1} r_{2} = \frac{c}{a}$ by r2 to express r1 in terms of r2.

$\frac{r_{1} r_{2}}{r_{2}} = \frac{c}{a r_{2}} r_{1} = \frac{c}{a r_{2}}$

Step 3. Calculations

Substitute $r_{1} = \frac{c}{a r_{2}}$ in $r_{1} + r_{2} = - \frac{b}{a}$ we get

role="math" localid="1646997829971" $\frac{c}{a r_{2}} + r_{2} = - \frac{b}{a} a r_{2} (\frac{c}{a r_{2}} + r_{2}) = - \frac{b}{a} \cdot a r_{2} c + a r_{2}^{2} = - b r_{2} a r_{2}^{2} + b r_{2} + c = 0 r_{2} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Step 4. r2 in terms of r1

Divide the equation $r_{1} r_{2} = \frac{c}{a}$ by r1 to express r2 in terms of r1.

$\frac{r_{1} r_{2}}{r_{1}} = \frac{c}{a r_{1}} r_{2} = \frac{c}{a r_{1}}$

Substitute $r_{2} = \frac{c}{a r_{1}}$ in $r_{1} + r_{2} = - \frac{b}{a}$ we get

$\frac{c}{a r_{1}} + r_{1} = - \frac{b}{a} c + a r_{1}^{2} = - b r_{1} a r_{1}^{2} + b r_{1} + c = 0 r_{1} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Step 5. r1 and r2

$r_{1} = \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} r_{2} = \frac{- b - \sqrt{b^{2} - 4 a c}}{2 a}$

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If r1 and r2 are two solutions of a quadratic equation ax2 + bx + c = 0, it can be shown that

$r_{1} + r_{2} = - \frac{b}{a} a n d r_{1} r_{2} = \frac{c}{a}$
Solve this system of equations for r1 and r2.

Short Answer

Step by step solution

Step 1. Given information

Step 2. r1 in terms of r2

Step 3. Calculations

Step 4. r2 in terms of r1

Step 5. r1 and r2

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If r1 and r2 are two solutions of a quadratic equation ax2 + bx + c = 0, it can be shown thatr1+r2=-baandr1r2=caSolve this system of equations for r1 and r2.

Short Answer

Step by step solution

Step 1. Given information

Step 2. r1 in terms of r2

Step 3. Calculations

Step 4. r2 in terms of r1

Step 5. r1 and r2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

If r1 and r2 are two solutions of a quadratic equation ax2 + bx + c = 0, it can be shown that

$r_{1} + r_{2} = - \frac{b}{a} a n d r_{1} r_{2} = \frac{c}{a}$
Solve this system of equations for r1 and r2.