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Chapter 3: Problem 3

Determine the values of \(x\) for which \(f(x)=0\). \(f(x)=-4 x^{2}+18 x+10\)

Short Answer

Expert verified
The values of \(x\) are \(-0.5\) and \5\.

Step by step solution

01

Identify the quadratic equation

The given function is a quadratic equation of the form \(ax^2 + bx + c = 0\). Here, \(f(x) = -4x^2 + 18x + 10\).
02

Write down the quadratic formula

The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = -4, b = 18, \text{and} c = 10\).
03

Calculate the discriminant

The discriminant is \(b^2 - 4ac\). Substitute the values: \(18^2 - 4(-4)(10) = 324 + 160 = 484\).
04

Apply the quadratic formula

Substitute the values into the quadratic formula: \(x = \frac{-18 \pm \sqrt{484}}{2(-4)} = \frac{-18 \pm 22}{-8}\).
05

Solve for the two values of x

Calculate the two potential solutions: \(x_1 = \frac{-18 + 22}{-8} = -0.5\) and \(x_2 = \frac{-18 - 22}{-8} = 5\).

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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 to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). To find the roots (solutions) of a quadratic equation, you can use the quadratic formula:
\(x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a}\).
This formula works for any quadratic equation, even if the roots are complex or irrational.
Here鈥檚 how to apply it:
  • Identify the coefficients \(a\), \(b\), and \(c\) in your equation.
  • Calculate the discriminant (\(b^2 - 4ac\)).
  • Substitute \(a\), \(b\), and the discriminant into the formula.
This method ensures you find the correct roots, whether one, two, or none.
discriminant
The discriminant is a key part of the quadratic formula, and it tells you about the nature of the roots of the equation. It is given by the expression \(b^2 - 4ac\).
The value of the discriminant determines the type of solutions you get:
  • If \(b^2 - 4ac > 0\), there are two distinct real roots.
  • If \(b^2 - 4ac = 0\), there is exactly one real root (also called a repeated root).
  • If \(b^2 - 4ac < 0\), the equation has no real roots but two complex roots.
In our example, where \(b = 18\) and \(a = -4\), and \(c = 10\), we calculated the discriminant as 484, which is greater than zero. This means our quadratic equation has two distinct real roots.
roots of a quadratic equation
The roots (or solutions) of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). Using the quadratic formula, we can find these roots by solving the formula:
\(x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a}\).
From the previous steps:
  • We calculated the discriminant as 484.
  • We then substituted into the quadratic formula: \(x = \frac{-18 \, \pm \, \sqrt{484}}{2(-4)}\).
  • Simplifying the above, we get two values for \(x\): \(x_1 = -0.5\) and \(x_2 = 5\).
These values are the roots of the quadratic equation \(-4x^2 + 18x + 10 = 0\). They are the points where the graph of the function \(f(x)\) crosses the x-axis.

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Most popular questions from this chapter

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ \sqrt{5-x}-7 \geq 0 $$

Given \(y=f(x)\) a. Divide the numerator by the denominator to write \(f(x)\) in the form \(f(x)=\) quotient \(+\frac{\text { remainder }}{\text { divisor }}\). b. Use transformations of \(y=\frac{1}{x}\) to graph the function. $$ f(x)=\frac{5 x+11}{x+2} $$

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ \left|x^{2}-4\right|<5 $$

For a certain stretch of road, the distance \(d\) (in \(\mathrm{ft}\) ) required to stop a car that is traveling at speed \(y\) (in mph) before the brakes are applied can be approximated by \(d(v)=0.06 v^{2}+2 v .\) Find the speeds for which the car can be stopped within \(250 \mathrm{ft}\).

The number of U.S. citizens of voting age, \(N(t)\) (in millions), can be modeled according to the number of years \(t\) since 1932 . $$N(t)=0.0135 t^{2}+1.09 t+73.2$$ The function defined by $$V(t)=1.08 t+36.9$$ represents the number of people who voted \(V(t)\) (in millions) in U.S. presidential elections ( \(t\) is the number of years since 1932 and \(t\) is a multiple of 4). (Source: U.S. Census Bureau, www.census.gov) a. Write the function defined by \(P(t)=\left(\frac{V}{N}\right)(t)\) and interpret its meaning in the context of this problem. b. Evaluate \(P(60)\) and interpret its meaning in the context of this problem. Round to 2 decimal places.
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