Question

solve the equation: $x^2 + 4x - 9 = 0$. fully simplify all answers, including non-real solutions.
$x = $
question help: video 1 video 2
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question 8 0/4 pts 3 99 details
use the quadratic formula to solve the equation $16x^2 + 56x + 65 = 0$.
enter multiple answers as a list separated by commas. example: $2 + 2i, 2 - 2i$
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question 9 0/4 pts 3 99 details
use the quadratic formula to solve the equation $x^2 + 2x + 5 = 0$.
enter multiple answers as a list separated by commas. example: $2 + 2i, 2 - 2i$
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question 10 0/4 pts 3 99 details
solve equation by the quadratic formula. list the solutions, separated by commas.
$2t^2 - t + 11 = 9$
$t = $
question help: video

Explanation:

Question 1:

Step1: Identify coefficients

For $x^2 + 4x - 9 = 0$, $a=1$, $b=4$, $c=-9$

Step2: Apply quadratic formula

Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Expression: $x=\frac{-4\pm\sqrt{4^2-4(1)(-9)}}{2(1)}$

Step3: Simplify discriminant

Calculate $b^2-4ac$: $\sqrt{16+36}=\sqrt{52}=2\sqrt{13}$

Step4: Final simplification

Expression: $x=\frac{-4\pm2\sqrt{13}}{2}=-2\pm\sqrt{13}$

Question 8:

Step1: Identify coefficients

For $16x^2 + 56x + 65 = 0$, $a=16$, $b=56$, $c=65$

Step2: Apply quadratic formula

Expression: $x=\frac{-56\pm\sqrt{56^2-4(16)(65)}}{2(16)}$

Step3: Simplify discriminant

Calculate $b^2-4ac$: $\sqrt{3136-4160}=\sqrt{-1024}=32i$

Step4: Final simplification

Expression: $x=\frac{-56\pm32i}{32}=-\frac{7}{4}\pm i$

Question 9:

Step1: Identify coefficients

For $x^2 + 2x + 5 = 0$, $a=1$, $b=2$, $c=5$

Step2: Apply quadratic formula

Expression: $x=\frac{-2\pm\sqrt{2^2-4(1)(5)}}{2(1)}$

Step3: Simplify discriminant

Calculate $b^2-4ac$: $\sqrt{4-20}=\sqrt{-16}=4i$

Step4: Final simplification

Expression: $x=\frac{-2\pm4i}{2}=-1\pm2i$

Question 10:

Step1: Rewrite in standard form

$2t^2 - t + 11 = 9$ simplifies to $2t^2 - t + 2 = 0$; $a=2$, $b=-1$, $c=2$

Step2: Apply quadratic formula

Expression: $t=\frac{-(-1)\pm\sqrt{(-1)^2-4(2)(2)}}{2(2)}$

Step3: Simplify discriminant

Calculate $b^2-4ac$: $\sqrt{1-16}=\sqrt{-15}=i\sqrt{15}$

Step4: Final simplification

Expression: $t=\frac{1\pm i\sqrt{15}}{4}$

Answer:

1. $x=-2+\sqrt{13}, -2-\sqrt{13}$
2. $x=-\frac{7}{4}+i, -\frac{7}{4}-i$
3. $x=-1+2i, -1-2i$
4. $t=\frac{1+i\sqrt{15}}{4}, \frac{1-i\sqrt{15}}{4}$