Problem 44 - Algebra 2

Question

Solve each equation by factoring, by taking square roots, or by graphing. If necessary, round your answer to the nearest hundredth. $$ 3 x^{2}+7 x=9 $$

Step-by-Step Solution

Answer

The solutions are approximately \(x \approx 0.92\) and \(x \approx -3.25\), rounded to the nearest hundredth.

Step 1: Move all terms to one side to set the equation to zero

Subtract 9 from both sides of the equation to set the quadratic equation to equal zero. This gives us the equation: \(3x^2 + 7x - 9 = 0\).

Step 2: Factor the quadratic equation

Since this quadratic does not factor easily, we can use the quadratic formula to find the roots. The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\). For our equation, \(a = 3\), \(b = 7\), and \(c = -9\).

Step 3: Apply the quadratic formula

Plugging the values into the quadratic formula gives us \(x = \frac{-7 \pm \sqrt{7^2 - 4(3)(-9)}}{2(3)}\).

Step 4: Simplify under the square root

Calculate the discriminant \(\Delta = b^2 - 4ac\), which is \(7^2 - 4(3)(-9) = 49 + 108 = 157\).

Step 5: Solve for x

Now we have \(x = \frac{-7 \pm \sqrt{157}}{6}\). There are two solutions for \(x\), which are \(x = \frac{-7 + \sqrt{157}}{6}\) and \(x = \frac{-7 - \sqrt{157}}{6}\).

Step 6: Approximate the solutions

Using a calculator, we can approximate the two solutions to \(x \approx \frac{-7 + 12.53}{6}\) and \(x \approx \frac{-7 - 12.53}{6}\), which simplifies to approximately \(x \approx 0.92\) and \(x \approx -3.25\), respectively. These values are rounded to the nearest hundredth.

Key Concepts

Factoring Quadratics

When solving quadratic equations, factoring is a powerful method that can simplify the problem significantly. To factor quadratics, one must rewrite the quadratic equation in the form of \( ax^2 + bx + c = 0 \) as a product of two binomials. The goal is to find two numbers that multiply to give the product \( ac \) and add up to \( b \). However, some quadratics, like \( 3x^2 + 7x - 9 \), resist easy factoring. In these cases, alternative methods such as the quadratic formula become essential.

Quadratic Formula

If a quadratic equation does not factor neatly or at all, the quadratic formula is a reliable tool to find the roots. The formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) can be applied for any quadratic equation \( ax^2 + bx + c = 0 \). The symbols \( \pm \) indicate that there will usually be two solutions to the equation. The quadratic formula provides an exact answer or an approximate value when necessary, as in the given equation, where it resulted in \( x \approx 0.92 \) and \( x \approx -3.25 \), rounded to the nearest hundredth.

Discriminant of a Quadratic Equation

The discriminant, denoted by \( \Delta \) and calculated as \( b^2 - 4ac \), gives us crucial information about the nature of the roots of a quadratic equation. A positive discriminant indicates two real and distinct solutions, a discriminant of zero corresponds to exactly one real solution, and a negative discriminant suggests the solutions are not real numbers but complex or imaginary. In our example, the discriminant is \( 157 \), which is positive, confirming the existence of two distinct real solutions.

Taking Square Roots

In some cases, taking square roots can be used to solve a quadratic equation, especially when it is in the form \( ax^2 = c \). You would isolate \( x^2 \) on one side and then take the square root of both sides. Remember to consider both the positive and negative roots. Nonetheless, this method wasn't applicable to the given equation, \( 3x^2 + 7x - 9 = 0 \), because it is not in an appropriate form for taking square roots directly.

Graphing Quadratic Functions

Graphing the corresponding quadratic function \( y = ax^2 + bx + c \) can provide a visual representation of the solutions to the quadratic equation. The points where the parabola intersects the x-axis represent the roots or solutions. This method enables an understanding of the behavior of the quadratic function, including its direction (upward for \( a > 0 \) and downward for \( a < 0 \) ) and its vertex, which is the highest or lowest point on the graph. Although not used here, graphing can be particularly helpful when seeking approximate solutions or analyzing the function's properties.