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Magazine Chapter 11: Problem 40
Let \(f(x)=6 x^{2}-7 x-20 .\) Find \(x\) such that \(f(x)=0\)
Short Answer
Expert verified
The values of \(x\) are 2.5 and -1.333.
Step by step solution
01
Understand the Problem
We are given the quadratic function \(f(x) = 6x^2 - 7x - 20\) and we need to find the value(s) of \(x\) such that \(f(x) = 0\).
02
Set the Equation to Zero
Set the function equal to zero to form a quadratic equation: \(6x^2 - 7x - 20 = 0\).
03
Identify Coefficients
Identify the coefficients in the quadratic equation \(ax^2 + bx + c = 0\). Here, \(a = 6\), \(b = -7\), and \(c = -20\).
04
Use the Quadratic Formula
Apply the quadratic formula to solve for \(x\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Substitute the values of \(a\), \(b\), and \(c\):
05
Calculate the Discriminant
First, calculate the discriminant \(b^2 - 4ac\): \[ (-7)^2 - 4(6)(-20) = 49 + 480 = 529 \].
06
Compute the Square Root of the Discriminant
Take the square root of 529: \( \sqrt{529} = 23 \).
07
Solve for x
Substitute the discriminant and solve for \(x\): \[ x = \frac{7 \pm 23}{12} \]. This gives us the solutions: \[ x = \frac{7 + 23}{12} = 2.5 \] and \[ x = \frac{7 - 23}{12} = -1.333 \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form: \[ ax^2 + bx + c = 0 \]It allows us to find the values of x that satisfy the equation. The formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, *a*, *b*, and *c* are the coefficients of the quadratic equation. By substituting the values of *a*, *b*, and *c* into the formula, we can find the solutions for *x*. This formula works for any quadratic equation, whether it has real or complex solutions.
Discriminant
The discriminant is a key part of the quadratic formula. It is represented by \( b^2 - 4ac \). The value of the discriminant tells us about the nature of the roots of the quadratic equation:
- If the discriminant is positive, the equation has two distinct real roots.
- If the discriminant is zero, the equation has exactly one real root (a repeated root).
- If the discriminant is negative, the equation has two complex roots.
Roots of Quadratic Equations
The roots of a quadratic equation are the values of *x* that make the equation true (i.e., the values of *x* that satisfy the equation \( ax^2 + bx + c = 0 \)). These roots can be found using the quadratic formula. After calculating the discriminant and identifying it as positive, we continue to solve for *x*: \[ x = \frac{7 \pm 23}{12} \] This gives us: \[ x_1 = \frac{7 + 23}{12} = 2.5 \] \[ x_2 = \frac{7 - 23}{12} = -1.333 \] So, the roots of the quadratic equation \( 6x^2 - 7x - 20 \) are 2.5 and -1.333. These are the values of *x* that make the function *f(x)* equal to zero.
Coefficients
Coefficients are the numerical values that multiply the variable terms in a quadratic equation. In the standard form of a quadratic equation \( ax^2 + bx + c = 0 \), *a*, *b*, and *c* are the coefficients. Each coefficient plays a significant role:
- *a* determines the curvature or the 'opening' direction of the parabola. If *a* is positive, the parabola opens upwards; if negative, it opens downwards.
- *b* influences the position of the vertex of the parabola along the x-axis.
- *c* is the constant term and determines the point at which the parabola intersects the y-axis.
