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Chapter 5: Problem 58

Solve. Let \(g(x)=2 x^{2}-15 x .\) Find \(a\) such that \(g(a)=-7\)

Short Answer

Expert verified
The values of \(a\) are 7 and \(\frac{1}{2}\).

Step by step solution

01

- Write down the given function

The given function is: \(g(x) = 2x^2 - 15x\)
02

- Set the function equal to -7

We need to find the value of \(a\) such that \(g(a) = -7\). Therefore, set the function equal to -7: \(2a^2 - 15a = -7\)
03

- Rearrange the equation

Move -7 to the left side of the equation to set the equation to 0: \(2a^2 - 15a + 7 = 0\)
04

- Solve the quadratic equation using the quadratic formula

The quadratic formula is given by: \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 2\), \(b = -15\), and \(c = 7\). Substitute these values into the quadratic formula.
05

- Compute the discriminant

Calculate the discriminant: \(b^2 - 4ac = (-15)^2 - 4 \cdot 2 \cdot 7\) \(= 225 - 56\) \(= 169\)
06

- Solve for a

Using the quadratic formula: \(a = \frac{-(-15) \pm \sqrt{169}}{2 \cdot 2}\) \(a = \frac{15 \pm 13}{4}\) This gives us two potential solutions: \(a = \frac{15 + 13}{4} = 7\) and \(a = \frac{15 - 13}{4} = \frac{1}{2}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic formula
Here鈥檚 how you can apply the quadratic formula step by step:
  • Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
  • Calculate the discriminant \(\Delta = b^2 - 4ac\).
  • Substitute \(\Delta\) into the formula along with \(a\) and \(b\).
  • Solve for the values of \(x\).

In our exercise, \(a = 2\), \(b = -15\), and \(c = 7\). Using these in the formula will help us solve for \(a\).
discriminant
Here鈥檚 how to understand what the discriminant means:
  • If \(\Delta > 0\), there are two distinct real roots.
  • If \(\Delta = 0\), there is exactly one real root.
  • If \(\Delta < 0\), there are no real roots, but two complex roots.

In the given exercise, \(\Delta\) is 169, which is positive, meaning our equation has two distinct real roots.
setting function equal to value
  • Substitute the given value into the function.
  • Set the equation equal to that value.
  • Rearrange the resulting equation to solve for the unknown variable.

This approach helps simplify many mathematical problems by providing a clear goal to reach.
rearranging equations
  • Identify the terms that need to be moved.
  • Use addition, subtraction, multiplication, or division to move terms across the equals sign.
  • Simplify the equation so that you have a standard form.

In our exercise, moving \(-7\) to the left gave us \(2a^2 - 15a + 7 = 0\). This standard form is necessary for applying the quadratic formula.

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