Solved on Dec 20, 2023

Solve the quadratic equation $2x^2 - 14x = -20$ by graphing the corresponding function.

STEP 1

Assumptions
1. We are given a quadratic equation $2x^2 - 14x = -20$ .
2. We need to solve this equation by graphing the corresponding function.
3. To graph the function, we will first convert the equation into standard form $y = ax^2 + bx + c$ .

STEP 2

First, we need to move all terms to one side of the equation to get it into the standard form.
$2x^2 - 14x + 20 = 0$

STEP 3

Now, we identify the coefficients for the quadratic equation in standard form, where $a = 2$ , $b = -14$ , and $c = 20$ .

STEP 4

The corresponding function to graph is $y = 2x^2 - 14x + 20$ .

STEP 5

To graph the function, we will find the vertex of the parabola. The x-coordinate of the vertex is given by $-\frac{b}{2a}$ .
$x_{vertex} = -\frac{-14}{2 \cdot 2}$

STEP 6

Calculate the x-coordinate of the vertex.
$x_{vertex} = -\frac{-14}{4} = \frac{14}{4} = 3.5$

STEP 7

Now, we will find the y-coordinate of the vertex by plugging $x_{vertex}$ into the function $y = 2x^2 - 14x + 20$ .
$y_{vertex} = 2(3.5)^2 - 14(3.5) + 20$

STEP 8

Calculate the y-coordinate of the vertex.
$y_{vertex} = 2(12.25) - 49 + 20$

STEP 9

Simplify the calculation for the y-coordinate.
$y_{vertex} = 24.5 - 49 + 20$

STEP 10

Finish calculating the y-coordinate of the vertex.
$y_{vertex} = -4.5$

STEP 11

The vertex of the parabola is at the point $(3.5, -4.5)$ .

STEP 12

Next, we will find the y-intercept of the parabola by setting $x = 0$ in the function $y = 2x^2 - 14x + 20$ .
$y_{intercept} = 2(0)^2 - 14(0) + 20$

STEP 13

Calculate the y-intercept.
$y_{intercept} = 0 - 0 + 20 = 20$

STEP 14

The y-intercept of the parabola is at the point $(0, 20)$ .

STEP 15

To find the x-intercepts (the solutions to the equation), we set $y = 0$ and solve for $x$ .
$0 = 2x^2 - 14x + 20$

STEP 16

We can solve for the x-intercepts by factoring the quadratic equation, if possible, or using the quadratic formula. Let's try factoring first.

STEP 17

We look for two numbers that multiply to $2 \cdot 20 = 40$ and add up to $-14$ . These numbers are $-10$ and $-4$ .

STEP 18

Factor the quadratic equation using these numbers.
$0 = (2x - 10)(x - 4)$

STEP 19

Set each factor equal to zero and solve for $x$ .
$2x - 10 = 0 \quad \text{or} \quad x - 4 = 0$

STEP 20

Solve the first equation for $x$ .
$2x = 10$
$x = 5$

STEP 21

Solve the second equation for $x$ .
$x = 4$

STEP 22

The x-intercepts of the parabola are at the points $(5, 0)$ and $(4, 0)$ .

STEP 23

Now we have enough points to sketch the graph of the parabola: the vertex $(3.5, -4.5)$ , the y-intercept $(0, 20)$ , and the x-intercepts $(5, 0)$ and $(4, 0)$ .

STEP 24

Plot these points on a coordinate plane and draw a smooth curve through them to represent the parabola.

STEP 25

The points where the parabola crosses the x-axis represent the solutions to the original equation.

STEP 26

The solutions to the equation $2x^2 - 14x = -20$ by graphing are $x = 5$ and $x = 4$ .
The solutions to the equation by graphing are $x = 5$ and $x = 4$ .

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Solve the quadratic equation 2x2−14x=−202x^2 - 14x = -202x2−14x=−20 by graphing the corresponding function.

STEP 1

Assumptions1. We are given a quadratic equation 2x2−14x=−202x^2 - 14x = -202x2−14x=−20.2. We need to solve this equation by graphing the corresponding function.3. To graph the function, we will first convert the equation into standard form y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c.

STEP 2

First, we need to move all terms to one side of the equation to get it into the standard form.2x2−14x+20=02x^2 - 14x + 20 = 02x2−14x+20=0

STEP 3

Now, we identify the coefficients for the quadratic equation in standard form, where a=2a = 2a=2, b=−14b = -14b=−14, and c=20c = 20c=20.

STEP 4

The corresponding function to graph is y=2x2−14x+20y = 2x^2 - 14x + 20y=2x2−14x+20.

STEP 5

To graph the function, we will find the vertex of the parabola. The x-coordinate of the vertex is given by −b2a-\frac{b}{2a}−2ab​.xvertex=−−142⋅2x_{vertex} = -\frac{-14}{2 \cdot 2}xvertex​=−2⋅2−14​

STEP 6

Calculate the x-coordinate of the vertex.xvertex=−−144=144=3.5x_{vertex} = -\frac{-14}{4} = \frac{14}{4} = 3.5xvertex​=−4−14​=414​=3.5

STEP 7

Now, we will find the y-coordinate of the vertex by plugging xvertexx_{vertex}xvertex​ into the function y=2x2−14x+20y = 2x^2 - 14x + 20y=2x2−14x+20.yvertex=2(3.5)2−14(3.5)+20y_{vertex} = 2(3.5)^2 - 14(3.5) + 20yvertex​=2(3.5)2−14(3.5)+20

STEP 8

Calculate the y-coordinate of the vertex.yvertex=2(12.25)−49+20y_{vertex} = 2(12.25) - 49 + 20yvertex​=2(12.25)−49+20

STEP 9

Simplify the calculation for the y-coordinate.yvertex=24.5−49+20y_{vertex} = 24.5 - 49 + 20yvertex​=24.5−49+20

STEP 10

Finish calculating the y-coordinate of the vertex.yvertex=−4.5y_{vertex} = -4.5yvertex​=−4.5

STEP 11

The vertex of the parabola is at the point (3.5,−4.5)(3.5, -4.5)(3.5,−4.5).

STEP 12

Next, we will find the y-intercept of the parabola by setting x=0x = 0x=0 in the function y=2x2−14x+20y = 2x^2 - 14x + 20y=2x2−14x+20.yintercept=2(0)2−14(0)+20y_{intercept} = 2(0)^2 - 14(0) + 20yintercept​=2(0)2−14(0)+20

STEP 13

Calculate the y-intercept.yintercept=0−0+20=20y_{intercept} = 0 - 0 + 20 = 20yintercept​=0−0+20=20

STEP 14

The y-intercept of the parabola is at the point (0,20)(0, 20)(0,20).

STEP 15

To find the x-intercepts (the solutions to the equation), we set y=0y = 0y=0 and solve for xxx.0=2x2−14x+200 = 2x^2 - 14x + 200=2x2−14x+20

STEP 16

We can solve for the x-intercepts by factoring the quadratic equation, if possible, or using the quadratic formula. Let's try factoring first.

STEP 17

We look for two numbers that multiply to 2⋅20=402 \cdot 20 = 402⋅20=40 and add up to −14-14−14. These numbers are −10-10−10 and −4-4−4.

STEP 18

Factor the quadratic equation using these numbers.0=(2x−10)(x−4)0 = (2x - 10)(x - 4)0=(2x−10)(x−4)

STEP 19

Set each factor equal to zero and solve for xxx.2x−10=0orx−4=02x - 10 = 0 \quad \text{or} \quad x - 4 = 02x−10=0orx−4=0

STEP 20

Solve the first equation for xxx.2x=102x = 102x=10x=5x = 5x=5

STEP 21

Solve the second equation for xxx.x=4x = 4x=4

STEP 22

The x-intercepts of the parabola are at the points (5,0)(5, 0)(5,0) and (4,0)(4, 0)(4,0).

STEP 23

Now we have enough points to sketch the graph of the parabola: the vertex (3.5,−4.5)(3.5, -4.5)(3.5,−4.5), the y-intercept (0,20)(0, 20)(0,20), and the x-intercepts (5,0)(5, 0)(5,0) and (4,0)(4, 0)(4,0).

STEP 24

Plot these points on a coordinate plane and draw a smooth curve through them to represent the parabola.

STEP 25

The points where the parabola crosses the x-axis represent the solutions to the original equation.

STEP 26

The solutions to the equation 2x2−14x=−202x^2 - 14x = -202x2−14x=−20 by graphing are x=5x = 5x=5 and x=4x = 4x=4.The solutions to the equation by graphing are x=5x = 5x=5 and x=4x = 4x=4.

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Solve the quadratic equation $2x^2 - 14x = -20$ by graphing the corresponding function.

Assumptions
1. We are given a quadratic equation $2x^2 - 14x = -20$ .
2. We need to solve this equation by graphing the corresponding function.
3. To graph the function, we will first convert the equation into standard form $y = ax^2 + bx + c$ .

First, we need to move all terms to one side of the equation to get it into the standard form.
$2x^2 - 14x + 20 = 0$

Now, we identify the coefficients for the quadratic equation in standard form, where $a = 2$ , $b = -14$ , and $c = 20$ .

The corresponding function to graph is $y = 2x^2 - 14x + 20$ .

To graph the function, we will find the vertex of the parabola. The x-coordinate of the vertex is given by $-\frac{b}{2a}$ .
$x_{vertex} = -\frac{-14}{2 \cdot 2}$

Calculate the x-coordinate of the vertex.
$x_{vertex} = -\frac{-14}{4} = \frac{14}{4} = 3.5$

Now, we will find the y-coordinate of the vertex by plugging $x_{vertex}$ into the function $y = 2x^2 - 14x + 20$ .
$y_{vertex} = 2(3.5)^2 - 14(3.5) + 20$

Calculate the y-coordinate of the vertex.
$y_{vertex} = 2(12.25) - 49 + 20$

Simplify the calculation for the y-coordinate.
$y_{vertex} = 24.5 - 49 + 20$

Finish calculating the y-coordinate of the vertex.
$y_{vertex} = -4.5$

The vertex of the parabola is at the point $(3.5, -4.5)$ .

Next, we will find the y-intercept of the parabola by setting $x = 0$ in the function $y = 2x^2 - 14x + 20$ .
$y_{intercept} = 2(0)^2 - 14(0) + 20$

Calculate the y-intercept.
$y_{intercept} = 0 - 0 + 20 = 20$

The y-intercept of the parabola is at the point $(0, 20)$ .

To find the x-intercepts (the solutions to the equation), we set $y = 0$ and solve for $x$ .
$0 = 2x^2 - 14x + 20$

We look for two numbers that multiply to $2 \cdot 20 = 40$ and add up to $-14$ . These numbers are $-10$ and $-4$ .

Factor the quadratic equation using these numbers.
$0 = (2x - 10)(x - 4)$

Set each factor equal to zero and solve for $x$ .
$2x - 10 = 0 \quad \text{or} \quad x - 4 = 0$

Solve the first equation for $x$ .
$2x = 10$
$x = 5$

Solve the second equation for $x$ .
$x = 4$

The x-intercepts of the parabola are at the points $(5, 0)$ and $(4, 0)$ .

Now we have enough points to sketch the graph of the parabola: the vertex $(3.5, -4.5)$ , the y-intercept $(0, 20)$ , and the x-intercepts $(5, 0)$ and $(4, 0)$ .

The solutions to the equation $2x^2 - 14x = -20$ by graphing are $x = 5$ and $x = 4$ .
The solutions to the equation by graphing are $x = 5$ and $x = 4$ .