Question

1. what type of solutions? (hint: use discriminant test)

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
a. $8x^2 + 8x + 10 = 8$
b. $7x^2 - 5x + 7 = -2$
c. $-5x^2 - 9x + 13 = 7$

Explanation:

For each quadratic equation, we first rewrite it in standard form $ax^2+bx+c=0$, then calculate the discriminant $\Delta = b^2-4ac$. The rules are:

• $\Delta>0$: Two distinct real solutions
• $\Delta=0$: One real repeated solution
• $\Delta<0$: Two complex conjugate solutions

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Part a

Step1: Rewrite to standard form

$8x^2 + 8x + 10 - 8 = 0$
$8x^2 + 8x + 2 = 0$

Step2: Identify $a,b,c$

$a=8$, $b=8$, $c=2$

Step3: Calculate discriminant

$\Delta = 8^2 - 4(8)(2) = 64 - 64 = 0$

Step4: Classify solutions

$\Delta=0$, so one real repeated solution

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Part b

Step1: Rewrite to standard form

$7x^2 - 5x + 7 + 2 = 0$
$7x^2 - 5x + 9 = 0$

Step2: Identify $a,b,c$

$a=7$, $b=-5$, $c=9$

Step3: Calculate discriminant

$\Delta = (-5)^2 - 4(7)(9) = 25 - 252 = -227$

Step4: Classify solutions

$\Delta<0$, so two complex conjugate solutions

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Part c

Step1: Rewrite to standard form

$-5x^2 - 9x + 13 - 7 = 0$
$-5x^2 - 9x + 6 = 0$

Step2: Identify $a,b,c$

$a=-5$, $b=-9$, $c=6$

Step3: Calculate discriminant

$\Delta = (-9)^2 - 4(-5)(6) = 81 + 120 = 201$

Step4: Classify solutions

$\Delta>0$, so two distinct real solutions

Answer:

a. One real repeated solution
b. Two complex conjugate solutions
c. Two distinct real solutions