State the solutions, or estimated solutions, to each quadratic equation in Exercises 23–26 by examining the graph of the related quadratic function equation 2x2 - 3x - 4 = 0 and Graph:y=2x^2-3x-4
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The graph shows the y axis has a positive 5 on the top and a negative one the bottom. The X axis show the same the same thing
y=3x^2-7
To find the solutions to the quadratic equation 2x^2 - 3x - 4 = 0 by examining the graph of the related quadratic function y = 2x^2 - 3x - 4, you need to look for the x-intercepts of the graph.
The x-intercepts are the points where the graph intersects the x-axis, which correspond to the solutions of the quadratic equation.
To determine the x-intercepts, set y = 0 in the quadratic function equation y = 2x^2 - 3x - 4 and solve for x:
0 = 2x^2 - 3x - 4
Now, you can use different techniques to solve this quadratic equation. One approach is to factor the expression, but in this case, the equation cannot be factored easily.
Another approach is to use the quadratic formula, which states that the solutions to a quadratic equation in the form ax^2 + bx + c = 0 can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the given equation 2x^2 - 3x - 4 = 0, you can identify the coefficients a = 2, b = -3, and c = -4.
Using the quadratic formula, you can plug in these values to find the solutions:
x = (-(3) ± √((-3)^2 - 4(2)(-4))) / (2(2))
= (-(-3) ± √(9 + 32)) / 4
= (3 ± √41) / 4
Therefore, the solutions to the quadratic equation 2x^2 - 3x - 4 = 0 are estimated to be:
x = (3 + √41) / 4
x = (3 - √41) / 4
These are the approximate values of x where the graph of y = 2x^2 - 3x - 4 intersects the x-axis, indicating the solutions to the quadratic equation.