Solve the following equation by graphing the related function.
1. 4x^2 + 8x = 32
A: ?
Use the Zero Product Property to solve the following equation.
2. x(x + 3) = 32
A: ?
To solve the first equation by graphing the related function, we'll graph the function y = 4x^2 + 8x - 32 and find the x-values where the graph intersects the x-axis (y = 0).
To begin, rewrite the equation in the form y = 4x^2 + 8x - 32:
4x^2 + 8x = 32
4x^2 + 8x - 32 = 0
Next, plot the graph of the equation y = 4x^2 + 8x - 32 on a coordinate plane. It will be a quadratic curve.
To find the x-intercepts (where the curve intersects the x-axis), locate the points on the graph where y = 0. These are the solutions to the equation.
Once you have graphed the equation and found the x-intercepts, the x-values where the graph intersects the x-axis will be the solutions to the equation. In this case, they are the values of x at the points where y = 0.
Now, let's move on to the second equation.
To solve the equation using the Zero Product Property, we'll set each factor of the equation equal to zero and solve for x.
The equation is x(x + 3) = 32.
To apply the Zero Product Property, we need to set each factor equal to zero:
x = 0 or x + 3 = 0
Now, solve each equation individually:
For the first equation, x = 0, there is a single solution, which is x = 0.
For the second equation, x + 3 = 0, we can isolate x by subtracting 3 from both sides of the equation:
x = -3
So the solutions to the equation x(x + 3) = 32 are x = 0 and x = -3.
To solve the equation 4x^2 + 8x = 32 by graphing, first rearrange the equation to the standard form: 4x^2 + 8x - 32 = 0.
To graph the related function, we need to plot the points where the graph intersects the x-axis.
Let's first evaluate the discriminant (b^2 - 4ac) of the quadratic equation:
Discriminant (D) = (8)^2 - 4(4)(-32)
= 64 + 512
= 576
Since the discriminant is positive (D > 0), the equation has two distinct real solutions.
Next, let's find the x-intercepts by factoring or using the quadratic formula.
To factor the equation, we can rewrite it as:
4(x^2 + 2x - 8) = 0
Now, we need to find two numbers that multiply to -8 and add up to 2.
Since 4 * (-2) = -8 and 4 - 2 = 2, the factors of the equation are (x - 2)(x + 4) = 0.
So, we have two solutions:
x - 2 = 0 => x = 2
x + 4 = 0 => x = -4
Now, we can plot the graph by determining the y-intercept and two x-intercepts on the coordinate plane.
The y-intercept is found by setting x = 0 and solving the equation:
4(0)^2 + 8(0) - 32 = 0
-32 = 0
So, the y-intercept is (0, -32).
The x-intercepts are (2, 0) and (-4, 0).
Now, we can plot these three points, connect them to form a parabola, and graph the related function.
The graph shows a downward-opening parabola that intersects the x-axis at x = 2 and x = -4.
To solve the equation x(x + 3) = 32 using the Zero Product Property, we need to set each factor equal to zero and solve for x.
So, we have the following two equations:
x = 0
x + 3 = 0
The first equation gives us the solution x = 0.
For the second equation, subtracting 3 from both sides gives:
x = -3
Therefore, the solutions to the equation x(x + 3) = 32 are x = 0 and x = -3.