go
find the x-intercepts for each of the quadratic equations.
16. $x^2 + 4x - 21 = 0$
17. $5x^2 + 16x + 3 = 0$
18. $x^2 - 4x + 1 = 0$
19. $x^2 - 16x + 4 = 0$
20. $x^2 + 3x - 40 = 0$
21. $2x^2 - 3x - 9 = 0$
22. $x^2 - 3x = 0$
23. $x^2 + 6x + 8 = 0$
24. $3x^2 - 11 = 0$

Question

Accepted Answer

### Problem 16: $ x^2 + 4x - 21 = 0 $
# Explanation:
## Step1: Factor the quadratic equation
We need two numbers that multiply to $-21$ and add up to $4$. The numbers are $7$ and $-3$ since $7\times(-3)=-21$ and $7 + (-3)=4$. So we can factor the equation as:
$ (x + 7)(x - 3) = 0 $
## Step2: Solve for $x$
Set each factor equal to zero:
$ x + 7 = 0 $ or $ x - 3 = 0 $
For $ x + 7 = 0 $, we get $ x = -7 $
For $ x - 3 = 0 $, we get $ x = 3 $
# Answer: $ x = -7 $ and $ x = 3 $

### Problem 17: $ 5x^2 + 16x + 3 = 0 $
# Explanation:
## Step1: Factor the quadratic equation
We need two numbers that multiply to $5\times3 = 15$ and add up to $16$. The numbers are $15$ and $1$. So we rewrite the middle term:
$ 5x^2 + 15x + x + 3 = 0 $
Factor by grouping:
$ 5x(x + 3) + 1(x + 3) = 0 $
$ (5x + 1)(x + 3) = 0 $
## Step2: Solve for $x$
Set each factor equal to zero:
$ 5x + 1 = 0 $ or $ x + 3 = 0 $
For $ 5x + 1 = 0 $, we get $ x = -\frac{1}{5} $
For $ x + 3 = 0 $, we get $ x = -3 $
# Answer: $ x = -\frac{1}{5} $ and $ x = -3 $

### Problem 18: $ x^2 - 4x + 1 = 0 $
# Explanation:
## Step1: Use the quadratic formula
The quadratic formula is $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ for an equation $ ax^2 + bx + c = 0 $. Here, $ a = 1 $, $ b = -4 $, and $ c = 1 $
## Step2: Calculate the discriminant
$ \Delta = b^2 - 4ac = (-4)^2 - 4(1)(1) = 16 - 4 = 12 $
## Step3: Solve for $x$
$ x = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3} $
# Answer: $ x = 2 + \sqrt{3} $ and $ x = 2 - \sqrt{3} $

### Problem 19: $ x^2 - 16x + 4 = 0 $
# Explanation:
## Step1: Use the quadratic formula
Here, $ a = 1 $, $ b = -16 $, and $ c = 4 $
## Step2: Calculate the discriminant
$ \Delta = (-16)^2 - 4(1)(4) = 256 - 16 = 240 $
## Step3: Solve for $x$
$ x = \frac{16 \pm \sqrt{240}}{2} = \frac{16 \pm 4\sqrt{15}}{2} = 8 \pm 2\sqrt{15} $
# Answer: $ x = 8 + 2\sqrt{15} $ and $ x = 8 - 2\sqrt{15} $

### Problem 20: $ x^2 + 3x - 40 = 0 $
# Explanation:
## Step1: Factor the quadratic equation
We need two numbers that multiply to $-40$ and add up to $3$. The numbers are $8$ and $-5$ since $8\times(-5)=-40$ and $8 + (-5)=3$. So we can factor the equation as:
$ (x + 8)(x - 5) = 0 $
## Step2: Solve for $x$
Set each factor equal to zero:
$ x + 8 = 0 $ or $ x - 5 = 0 $
For $ x + 8 = 0 $, we get $ x = -8 $
For $ x - 5 = 0 $, we get $ x = 5 $
# Answer: $ x = -8 $ and $ x = 5 $

### Problem 21: $ 2x^2 - 3x - 9 = 0 $
# Explanation:
## Step1: Factor the quadratic equation
We need two numbers that multiply to $2\times(-9)=-18$ and add up to $-3$. The numbers are $-6$ and $3$. So we rewrite the middle term:
$ 2x^2 - 6x + 3x - 9 = 0 $
Factor by grouping:
$ 2x(x - 3) + 3(x - 3) = 0 $
$ (2x + 3)(x - 3) = 0 $
## Step2: Solve for $x$
Set each factor equal to zero:
$ 2x + 3 = 0 $ or $ x - 3 = 0 $
For $ 2x + 3 = 0 $, we get $ x = -\frac{3}{2} $
For $ x - 3 = 0 $, we get $ x = 3 $
# Answer: $ x = -\frac{3}{2} $ and $ x = 3 $

### Problem 22: $ x^2 - 3x = 0 $
# Explanation:
## Step1: Factor the quadratic equation
Factor out $x$:
$ x(x - 3) = 0 $
## Step2: Solve for $x$
Set each factor equal to zero:
$ x = 0 $ or $ x - 3 = 0 $
For $ x - 3 = 0 $, we get $ x = 3 $
# Answer: $ x = 0 $ and $ x = 3 $

### Problem 23: $ x^2 + 6x + 8 = 0 $
# Explanation:
## Step1: Factor the quadratic equation
We need two numbers that multiply to $8$ and add up to $6$. The numbers are $4$ and $2$ since $4\times2 = 8$ and $4 + 2 = 6$. So we can factor the equation as:
$ (x + 4)(x + 2) = 0 $
## Step2: Solve for $x$
Set each factor equal to zero:
$ x + 4 = 0 $ or $ x + 2 = 0 $
For $ x + 4 = 0 $, we get $ x = -4 $
For $ x + 2 = 0 $, we get $ x = -2 $
# Answer: $ x = -4 $ and $ x = -2 $

### Problem 24: $ 3x^2 - 11 = 0 $
# Explanation:
## Step1: Solve for $x^2$
Add $11$ to both sides:
$ 3x^2 = 11 $
Divide both sides by $3$:
$ x^2 = \frac{11}{3} $
## Step2: Solve for $x$
Take the square root of both sides:
$ x = \pm\sqrt{\frac{11}{3}} = \pm\frac{\sqrt{33}}{3} $
# Answer: $ x = \frac{\sqrt{33}}{3} $ and $ x = -\frac{\sqrt{33}}{3} $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 16: \( x^2 + 4x - 21 = 0 \)

Step1: Factor the quadratic equation

Step2: Solve for \(x\)

Step1: Factor the quadratic equation

Step2: Solve for \(x\)

Step1: Use the quadratic formula

Step2: Calculate the discriminant

Step3: Solve for \(x\)

Answer:

Problem 17: \( 5x^2 + 16x + 3 = 0 \)