Problem 68 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\left(3 x^{2}+52 x\right)^{1 / 4}=4$$

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x = -\frac{64}{3}\).

Step 1 - Isolate the exponentiated term

The given equation is \(\big(3x^2 + 52x\big)^{1/4} = 4\). Isolate the term inside the parentheses by eliminating the exponent by raising both sides of the equation to the power of 4.

Step 2 - Raise both sides to the power of 4

To eliminate the fourth root, raise both sides to the power of 4: \(\big(\big(3x^2 + 52x\big)^{1/4}\big)^4 = 4^4\). Simplify the equation to: \(3x^2 + 52x = 256\).

Step 3 - Rewrite as a standard quadratic equation

Subtract 256 from both sides to form a standard quadratic equation: \(3x^2 + 52x - 256 = 0\).

Step 4 - Solve the quadratic equation using the quadratic formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 3\), \(b = 52\), and \(c = -256\). Substitute these values into the formula: \(x = \frac{-52 \pm \sqrt{52^2 - 4(3)(-256)}}{2(3)}\).

Step 5 - Simplify the discriminant

Calculate the discriminant: \(52^2 - 4\cdot 3 \cdot (-256) = 2704 + 3072 = 5776\). So the expression simplifies to: \(x = \frac{-52 \pm \sqrt{5776}}{6}\).

Step 6 - Find the square root and solve for x

The square root of 5776 is 76. So, we have: \(x = \frac{-52 \pm 76}{6}\). This gives us two solutions for x: \(x = \frac{-52 + 76}{6} = 4\) and \(x = \frac{-52 - 76}{6} = -\frac{128}{6} = -\frac{64}{3}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). The formula itself is: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this equation,

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term.

By substituting the values of \(a\), \(b\), and \(c\) into the formula, we can find the solutions for \(x\). It is important to remember that the \(\pm\) symbol means that there are usually two possible solutions. For example, in our given problem \(3x^2 + 52x - 256 = 0\), we identify

• \(a = 3\)
• \(b = 52\)
• \(c = -256\).

Substituting these into the quadratic formula gives us: \(x = \frac{-52 \pm \sqrt{52^2 - 4 \cdot 3 \cdot (-256)}}{2 \cdot 3}\).

Isolating Terms

Isolating terms is a key step in solving equations, including quadratic equations. It involves rearranging the equation so that the term we are focusing on is alone on one side of the equation. In our problem \((3x^2 + 52x)^{1/4} = 4\), we need to isolate the term inside the parentheses. To do this, we eliminate the fourth root by raising both sides of the equation to the power of 4. This looks like: \((3x^2 + 52x)^{1/4})^4 = 4^4\). This simplifies to: \(3x^2 + 52x = 256\). This step is crucial because it allows us to work with a simpler, more manageable quadratic equation. Sometimes it may involve adding or subtracting terms from both sides of the equation, or multiplying or dividing both sides by a number.

Discriminant Analysis

The discriminant is the part of the quadratic formula under the square root symbol: \(b^2 - 4ac\). It tells us about the nature of the solutions of the quadratic equation.
Depending on the value of the discriminant, there are three possible scenarios:

• If the discriminant is positive, there are two real and distinct solutions.
• If it is zero, there is exactly one real solution (the solutions are the same).
• If it is negative, there are no real solutions; instead, there are two complex solutions.

In our example, the discriminant is: \(52^2 - 4 \cdot 3 \cdot (-256) = 2704 + 3072 = 5776\), which is positive, indicating that we have two real solutions. Performing the discriminant analysis helps us understand how many and what type of solutions we should expect.

Simplifying Equations

Simplifying equations makes them much easier to solve. In the context of quadratic equations, simplifying usually involves several steps, including isolating terms, moving constants to one side, and combining like terms.
For example, in our problem after isolating the exponentiated term, and rewriting as a standard quadratic equation, we arrived at \(3x^2 + 52x = 256\). By rearranging, we get: \(3x^2 + 52x - 256 = 0\), which is now in the form \(ax^2 + bx + c = 0\). This is a standard quadratic equation and can be solved using the quadratic formula.
After substituting into the quadratic formula and simplifying the discriminant, we further simplify to find: \(x = \frac{-52 \pm 76}{6}\). This finally leads to the solutions: \(x = 4\) and \(x = -\frac{64}{3}\). Thus, the steps of simplification help us break down complex problems into easier parts that we can solve step-by-step.