Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$16 x^{2}+24 x+9=0$$

Step-by-Step Solution

Answer

The solution to the quadratic equation is \(x_{1} = x_{2} = -0.75\).

Step 1: Write down the Quadratic Formula

The Quadratic Formula is given by \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\)

Step 2: Substitute the values into the formula

Substitute \(a = 16\), \(b = 24\), and \(c = 9\) into the formula to get: \(x = \frac{-24 \pm \sqrt{{24}^{2}-4*16*9}}{2*16}\)

Step 3: Simplify the expression

Solving the equation we get \(x = \frac{-24 \pm \sqrt{{576}-576}}{32}\), which simplifies to \(x = \frac{-24 \pm 0}{32}\)

Step 4: Solve for x

Solving for x gives us two solutions: \(x_{1} = \frac{-24 + 0}{32} = -0.75\) and \(x_{2} = \frac{-24 - 0}{32} = -0.75\). So, \(x_{1} = x_{2} = -0.75\).

Step 5: Verify solutions graphically

By plotting the quadratic equation \(16 x^{2}+24 x+9=0\), you should see that the parabola intersects the x-axis at \(x = -0.75\), confirming the solution.

Key Concepts

Quadratic Equations

Quadratic equations are polynomials of degree two. They usually take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown. This equation forms a parabola when graphed on the Cartesian coordinate plane.

The key to solving quadratic equations is identifying the coefficients \( a \), \( b \), and \( c \). For the example \( 16x^2 + 24x + 9 = 0 \), our values are \( a = 16 \), \( b = 24 \), and \( c = 9 \). Once identified, these coefficients can be substituted into the quadratic formula. The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) allows us to find the roots or solutions of the equation. These solutions indicate where the graph of the equation will intercept the x-axis.

Quadratic equations can have zero, one, or two real solutions. If the term under the square root \( b^2 - 4ac \) (known as the discriminant) is positive, there are two distinct solutions. If it is zero, both solutions are the same. If it is negative, the solutions are complex numbers.

Graphing Utility

Graphing utilities are tools that help visualize mathematical equations, especially useful for understanding quadratics. They allow students to plot equations easily, observe the shape of the curve, and verify solutions graphically.

Using a graphing utility involves inputting the quadratic equation and observing where the parabola crosses the x-axis, if at all. With the equation \(16x^{2}+24x+9=0\), you would expect to see the parabola touch the x-axis at \(x = -0.75\). Such visuals can reinforce algebraic solutions and provide a clear picture of the solutions' correctness.

Graphing utilities are available in various forms, like software applications, online graphing calculators, and even specialized calculators. For students, these tools can be invaluable for checking work and gaining a deeper understanding of the behavior of quadratic equations.

Verification of Solutions

Verification of solutions serves as a confirmation step to ensure that the solutions derived from algebraic manipulations are correct. In the context of quadratic equations, this can be done through substitution or graphing.

Once you have the solutions from the quadratic formula, substitute them back into the original equation to see if they truly satisfy it. For \( x = -0.75 \), substituting back into \( 16x^2 + 24x + 9 = 0 \) should yield zero, confirming the solution.

Graphical verification involves plotting the equation and visually checking the intercepts. As described, the parabola should intersect the x-axis at \( x = -0.75 \). This double-checks that the algebraic solution is valid.

• Algebraic verification through substitution
• Graphical verification using graphing utilities

Both methods are crucial in learning and ensure accuracy in solving quadratic equations.