Question

Solve x^2=5^20 by inspection. There are two real solutions. Enter the lesser number first and use exponents in your responses.

(1 point)

Answer 1

Solve the following quadratic equation using square roots:

−3+(x+12)^2/−2=−3

Type the smaller solution first.

Answer 2

What are the key points on the graph of y=x2−8x+15
𝑦
=
𝑥
2
−
8
𝑥
+
15
? Name the vertex, x-intercepts, and y-intercepts.

(1 point)
x-intercepts: (, ) and (5, 0)

y-intercept: (, )

vertex: (, )

Answer 3

Determine the zeroof the quadratic function y=8x2+14x+3
Show all steps.

Step 1: Plug the equation into my calculator.

Step 2: I will look for the

Answer 4

Determine the vertex and axis of symmetry of y=x2−4x+19


The vertex is (, ).

The axis is symmetry is ​​​​​​​.

Answer 5

To solve the equation \( x^2 = 5^{20} \) by inspection, we can take the square root of both sides.

Calculating this gives:

\[
x = \pm \sqrt{5^{20}}
\]

Using the property of exponents, we know that:

\[
\sqrt{5^{20}} = 5^{10}
\]

Thus, the two solutions are:

\[
x = 5^{10} \quad \text{and} \quad x = -5^{10}
\]

Since we are asked to enter the lesser number first, the solutions are:

\[
-5^{10}, \, 5^{10}
\]

So, the final answer is:

\[
-5^{10}, \, 5^{10}
\]

Answer 6

To solve the quadratic equation

\[
-3 + \frac{(x + 12)^2}{-2} = -3,
\]

we first simplify the equation. Start by isolating the term with the square:

\[
\frac{(x + 12)^2}{-2} = -3 + 3.
\]

This simplifies to:

\[
\frac{(x + 12)^2}{-2} = 0.
\]

Next, we multiply both sides by -2 to eliminate the fraction:

\[
(x + 12)^2 = 0.
\]

Now, take the square root of both sides:

\[
x + 12 = 0.
\]

Solving for \(x\) gives:

\[
x = -12.
\]

Since this is a quadratic equation that results in a single solution, we conclude that the solution is:

\[
-12.
\]

Given that there is only one solution, it is also the smaller solution. Thus, the final answer is:

\[
-12.
\]

Answer 7

To analyze the quadratic equation \( y = x^2 - 8x + 15 \), we will identify the key points: the vertex, x-intercepts, and y-intercept.

### 1. X-Intercepts
The x-intercepts occur where \( y = 0 \):

\[
0 = x^2 - 8x + 15.
\]

Factoring the quadratic, we look for two numbers that multiply to \( 15 \) and add to \( -8 \). These numbers are \( -3 \) and \( -5 \):

\[
0 = (x - 3)(x - 5).
\]

Setting each factor to zero gives:

\[
x - 3 = 0 \quad \Rightarrow \quad x = 3,
\]
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5.
\]

Thus, the x-intercepts are \( (3, 0) \) and \( (5, 0) \).

### 2. Y-Intercept
The y-intercept occurs where \( x = 0 \):

\[
y = 0^2 - 8(0) + 15 = 15.
\]

Thus, the y-intercept is \( (0, 15) \).

### 3. Vertex
The vertex of a quadratic \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:

\[
x = -\frac{b}{2a}.
\]
Here, \( a = 1 \) and \( b = -8 \):

\[
x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4.
\]

Now, substituting \( x = 4 \) back into the equation to find the y-coordinate of the vertex:

\[
y = (4)^2 - 8(4) + 15 = 16 - 32 + 15 = -1.
\]

Thus, the vertex is \( (4, -1) \).

### Summary of Key Points
- **X-Intercepts:** \( (3, 0) \) and \( (5, 0) \)
- **Y-Intercept:** \( (0, 15) \)
- **Vertex:** \( (4, -1) \)

### Final Response
- **x-intercepts:** \( (3, 0) \) and \( (5, 0) \)
- **y-intercept:** \( (0, 15) \)
- **vertex:** \( (4, -1) \)

Answer 8

To find the zeros (roots) of the quadratic function \( y = 8x^2 + 14x + 3 \), we can use the quadratic formula. The steps are as follows:

### Step 1: Understand the Quadratic Formula
The zeros of a quadratic equation \( ax^2 + bx + c = 0 \) can be found using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this case, we have:
- \( a = 8 \)
- \( b = 14 \)
- \( c = 3 \)

### Step 2: Calculate the Discriminant
Before applying the formula, we calculate the discriminant \( D \):

\[
D = b^2 - 4ac
\]

Substituting the values:

\[
D = (14)^2 - 4(8)(3)
\]
\[
D = 196 - 96
\]
\[
D = 100
\]

### Step 3: Apply the Quadratic Formula
Now we use the discriminant in the quadratic formula:

\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]
Substituting \( b \) and \( D \):

\[
x = \frac{-14 \pm \sqrt{100}}{2 \cdot 8}
\]
\[
x = \frac{-14 \pm 10}{16}
\]

### Step 4: Calculate the Two Roots
Now we calculate the two possible values for \( x \):

1. For the plus sign:

\[
x_1 = \frac{-14 + 10}{16} = \frac{-4}{16} = -\frac{1}{4}
\]

2. For the minus sign:

\[
x_2 = \frac{-14 - 10}{16} = \frac{-24}{16} = -\frac{3}{2}
\]

### Final Zeros of the Quadratic Function
The zeros of the quadratic function \( y = 8x^2 + 14x + 3 \) are:

\[
x_1 = -\frac{1}{4}, \quad x_2 = -\frac{3}{2}
\]

### Final Response
- **Zeros:** \( -\frac{1}{4} \) and \( -\frac{3}{2} \)

Answer 9

To determine the vertex and axis of symmetry of the quadratic function \( y = x^2 - 4x + 19 \), we can use the formulas related to the vertex of a parabola.

### Step 1: Axis of Symmetry
The axis of symmetry for a quadratic function in the form \( y = ax^2 + bx + c \) is given by the formula:

\[
x = -\frac{b}{2a}
\]

In this case, \( a = 1 \) and \( b = -4 \). Plugging in these values:

\[
x = -\frac{-4}{2(1)} = \frac{4}{2} = 2
\]

So, the axis of symmetry is:

**Axis of Symmetry:** \( x = 2 \)

### Step 2: Vertex
To find the vertex, we can substitute the x-coordinate of the vertex back into the original equation. The x-coordinate of the vertex is \( x = 2 \). Now, plug this value into the equation:

\[
y = (2)^2 - 4(2) + 19
\]
\[
y = 4 - 8 + 19
\]
\[
y = 15
\]

Thus, the vertex of the parabola is:

**Vertex:** \( (2, 15) \)

### Final Response:
- **Vertex:** \( (2, 15) \)
- **Axis of Symmetry:** \( x = 2 \)