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Magazine Chapter 10: Problem 72
Let \(g(x)=4.5 x^{2}+0.2 x .\) For what value(s) of \(x\) is \(g(x)=3.75 ?\)
Short Answer
Expert verified
The values of \(x\) that satisfy \(g(x) = 3.75\) are approximately 0.89 and -0.94.
Step by step solution
01
Set the Function Equal to the Given Value
We are given the function \(g(x) = 4.5x^2 + 0.2x\) and want to find when it is equal to 3.75. Start by setting the equation: \[ 4.5x^2 + 0.2x = 3.75 \]
02
Rearrange the Equation into a Standard Quadratic Form
Next, we rearrange the equation to match the standard quadratic form \(ax^2 + bx + c = 0\). Subtract 3.75 from both sides: \[ 4.5x^2 + 0.2x - 3.75 = 0 \]
03
Identify Coefficients
In the quadratic equation \(4.5x^2 + 0.2x - 3.75 = 0\), identify the coefficients: \(a = 4.5\), \(b = 0.2\), and \(c = -3.75\).
04
Use the Quadratic Formula
Apply the quadratic formula to find the values of \(x\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Plugging in the coefficients gives: \[ x = \frac{-0.2 \pm \sqrt{(0.2)^2 - 4(4.5)(-3.75)}}{2(4.5)} \]
05
Calculate the Discriminant
Calculate the discriminant \(b^2 - 4ac\) to find if there are real solutions: \[ (0.2)^2 - 4(4.5)(-3.75) = 0.04 + 67.5 = 67.54 \] Since the discriminant is positive, there are two real solutions.
06
Solve for \(x\)
Use the quadratic formula to find the two values of \(x\): \[ x = \frac{-0.2 \pm \sqrt{67.54}}{9} \] Calculate each part: \[ x_1 = \frac{-0.2 + 8.218}{9} \approx 0.89 \] \[ x_2 = \frac{-0.2 - 8.218}{9} \approx -0.94 \]
07
Conclusion
The values of \(x\) for which \(g(x) = 3.75\) are approximately \(x \approx 0.89\) and \(x \approx -0.94\).
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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 polynomials of the form \(ax^2 + bx + c = 0\). Often, these equations may not factor easily, or factoring might not be evident. This is where the quadratic formula shines.
The formula is:\[\ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
The formula is:\[\ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- The formula provides a straightforward method to find the \(x\)-intercepts or roots of a quadratic equation.
- By plugging in values for \(a\), \(b\), and \(c\) from your specific equation, you can solve for \(x\).
- The "\(\pm\)" symbol indicates that there can be two possible solutions: one where you add the square root and one where you subtract it.
Discriminant
The discriminant is a key part of the quadratic formula and it tells you the nature of the roots without solving the equation fully. The discriminant is the part under the square root: \(b^2 - 4ac\).
- If the discriminant is positive, \(b^2 - 4ac > 0\), there are two distinct real roots. This means the equation crosses the x-axis at two points.
- If the discriminant is zero, \(b^2 - 4ac = 0\), there is exactly one real root, implying the vertex of the parabola touches the x-axis.
- If the discriminant is negative, \(b^2 - 4ac < 0\), there are no real roots, and the graph does not intersect the x-axis. Instead, the roots are complex numbers.
Standard Form of Quadratic Equation
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero. Rearranging any quadratic expression into this form is crucial for using the quadratic formula.
- The coefficient \(a\) is the quadratic coefficient, \(b\) is the linear coefficient, and \(c\) is the constant term.
- To rewrite an equation in standard form, combine like terms and ensure that the equation is set equal to zero.
- Once in standard form, identify \(a\), \(b\), and \(c\) to simplify the process of applying the quadratic formula.
