Math / Algebra

QuestionGraph the functions on the same coordinate plane. $\begin{array}{l} f(x)=-2 x \\ g(x)=x^{2}-3 \end{array}$
What are the solutions to the equation $f(x)=g(x)$ ? Select each correct answer. There are two.
Select 2 correct answer(s) -2 0 1 $-3$

Studdy Solution

STEP 1

1. The functions $f(x) = -2x$ and $g(x) = x^2 - 3$ are defined for all real numbers.
2. The solutions to $f(x) = g(x)$ are the $x$ -values where the graphs of the functions intersect.

STEP 2

1. Graph the functions on the same coordinate plane.
2. Set the functions equal to each other to find the intersection points.
3. Solve the resulting equation for $x$ .
4. Verify the solutions by substitution.

STEP 3

Graph the linear function $f(x) = -2x$ . This is a straight line with a slope of $-2$ and a y-intercept at the origin $(0, 0)$ .

STEP 4

Graph the quadratic function $g(x) = x^2 - 3$ . This is a parabola opening upwards with its vertex at $(0, -3)$ .

STEP 5

Set the functions equal to each other to find the intersection points:
$-2x = x^2 - 3$

STEP 6

Rearrange the equation to form a standard quadratic equation. Add $2x$ to both sides:
$0 = x^2 + 2x - 3$

STEP 7

Factor the quadratic equation:
$x^2 + 2x - 3 = (x + 3)(x - 1) = 0$

STEP 8

Apply the zero product property to find the solutions:
$x + 3 = 0 \quad \text{or} \quad x - 1 = 0$

STEP 9

Solve for $x$ :
$x = -3 \quad \text{or} \quad x = 1$

STEP 10

Verify the solutions by substituting back into the original functions:
1. For $x = -3$ : - $f(-3) = -2(-3) = 6$ - $g(-3) = (-3)^2 - 3 = 9 - 3 = 6$ - Both functions equal 6, so $x = -3$ is a solution.
2. For $x = 1$ : - $f(1) = -2(1) = -2$ - $g(1) = 1^2 - 3 = 1 - 3 = -2$ - Both functions equal -2, so $x = 1$ is a solution.
The solutions to the equation $f(x) = g(x)$ are:
-3, 1

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QuestionGraph the functions on the same coordinate plane. $\begin{array}{l} f(x)=-2 x \\ g(x)=x^{2}-3 \end{array}$
What are the solutions to the equation $f(x)=g(x)$ ? Select each correct answer. There are two.
Select 2 correct answer(s) -2 0 1 $-3$

Studdy Solution

STEP 1

1. The functions $f(x) = -2x$ and $g(x) = x^2 - 3$ are defined for all real numbers.
2. The solutions to $f(x) = g(x)$ are the $x$ -values where the graphs of the functions intersect.

STEP 2

1. Graph the functions on the same coordinate plane.
2. Set the functions equal to each other to find the intersection points.
3. Solve the resulting equation for $x$ .
4. Verify the solutions by substitution.

STEP 3

Graph the linear function $f(x) = -2x$ . This is a straight line with a slope of $-2$ and a y-intercept at the origin $(0, 0)$ .

STEP 4

Graph the quadratic function $g(x) = x^2 - 3$ . This is a parabola opening upwards with its vertex at $(0, -3)$ .

STEP 5

Set the functions equal to each other to find the intersection points:
$-2x = x^2 - 3$

STEP 6

Rearrange the equation to form a standard quadratic equation. Add $2x$ to both sides:
$0 = x^2 + 2x - 3$

STEP 7

Factor the quadratic equation:
$x^2 + 2x - 3 = (x + 3)(x - 1) = 0$

STEP 8

Apply the zero product property to find the solutions:
$x + 3 = 0 \quad \text{or} \quad x - 1 = 0$

STEP 9

Solve for $x$ :
$x = -3 \quad \text{or} \quad x = 1$

STEP 10

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Studdy Solution

STEP 1

1. The functions f(x)=−2x f(x) = -2x f(x)=−2x and g(x)=x2−3 g(x) = x^2 - 3 g(x)=x2−3 are defined for all real numbers.2. The solutions to f(x)=g(x) f(x) = g(x) f(x)=g(x) are the x x x-values where the graphs of the functions intersect.

STEP 2

1. Graph the functions on the same coordinate plane.2. Set the functions equal to each other to find the intersection points.3. Solve the resulting equation for x x x.4. Verify the solutions by substitution.

STEP 3

Graph the linear function f(x)=−2x f(x) = -2x f(x)=−2x. This is a straight line with a slope of −2-2−2 and a y-intercept at the origin (0,0)(0, 0)(0,0).

STEP 4

Graph the quadratic function g(x)=x2−3 g(x) = x^2 - 3 g(x)=x2−3. This is a parabola opening upwards with its vertex at (0,−3)(0, -3)(0,−3).

STEP 5

Set the functions equal to each other to find the intersection points:−2x=x2−3 -2x = x^2 - 3 −2x=x2−3

STEP 6

Rearrange the equation to form a standard quadratic equation. Add 2x 2x 2x to both sides:0=x2+2x−3 0 = x^2 + 2x - 3 0=x2+2x−3

STEP 7

Factor the quadratic equation:x2+2x−3=(x+3)(x−1)=0 x^2 + 2x - 3 = (x + 3)(x - 1) = 0 x2+2x−3=(x+3)(x−1)=0

STEP 8

Apply the zero product property to find the solutions:x+3=0orx−1=0 x + 3 = 0 \quad \text{or} \quad x - 1 = 0 x+3=0orx−1=0

STEP 9

Solve for x x x:x=−3orx=1 x = -3 \quad \text{or} \quad x = 1 x=−3orx=1

STEP 10

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1. The functions $f(x) = -2x$ and $g(x) = x^2 - 3$ are defined for all real numbers.
2. The solutions to $f(x) = g(x)$ are the $x$ -values where the graphs of the functions intersect.

1. Graph the functions on the same coordinate plane.
2. Set the functions equal to each other to find the intersection points.
3. Solve the resulting equation for $x$ .
4. Verify the solutions by substitution.

Graph the linear function $f(x) = -2x$ . This is a straight line with a slope of $-2$ and a y-intercept at the origin $(0, 0)$ .

Graph the quadratic function $g(x) = x^2 - 3$ . This is a parabola opening upwards with its vertex at $(0, -3)$ .

Set the functions equal to each other to find the intersection points:
$-2x = x^2 - 3$

Rearrange the equation to form a standard quadratic equation. Add $2x$ to both sides:
$0 = x^2 + 2x - 3$

Factor the quadratic equation:
$x^2 + 2x - 3 = (x + 3)(x - 1) = 0$

Apply the zero product property to find the solutions:
$x + 3 = 0 \quad \text{or} \quad x - 1 = 0$

Solve for $x$ :
$x = -3 \quad \text{or} \quad x = 1$