Let's solve the quadratic equation 5x squared plus 6x minus 32 equals 0 using the quadratic formula. First, we need to identify the coefficients. In this equation, a equals 5, b equals 6, and c equals negative 32. Next, we'll apply the quadratic formula, which states that x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. The graph shows our quadratic function, which crosses the x-axis at two points - these will be our solutions.
Now, let's substitute our values into the quadratic formula. We have x equals negative 6 plus or minus the square root of 6 squared minus 4 times 5 times negative 32, all divided by 2 times 5. Next, we calculate the discriminant, which is 6 squared minus 4 times 5 times negative 32. That's 36 plus 640, which equals 676. Now we can simplify our expression to x equals negative 6 plus or minus the square root of 676, all divided by 10. Since the square root of 676 is 26, we get x equals negative 6 plus or minus 26, all divided by 10. On our graph, we can see these solutions where the parabola crosses the x-axis.
Now, let's calculate our two solutions. For x1, we have negative 6 plus 26, all divided by 10, which equals 20 divided by 10, or simply 2. For x2, we have negative 6 minus 26, all divided by 10, which equals negative 32 divided by 10, or negative 16 over 5. So our solutions are x equals 2 and x equals negative 16 over 5. Let's verify these solutions by substituting them back into the original equation. For x equals 2, we get 5 times 2 squared plus 6 times 2 minus 32, which equals 5 times 4 plus 12 minus 32, which equals 20 plus 12 minus 32, which equals 0. For x equals negative 16 over 5, after calculating, we also get 0. So both solutions check out, confirming our answers.
To summarize what we've learned: The quadratic formula is a powerful tool that allows us to solve any quadratic equation in the form ax squared plus bx plus c equals zero. For our equation 5x squared plus 6x minus 32 equals zero, we identified the coefficients as a equals 5, b equals 6, and c equals negative 32. The discriminant, b squared minus 4ac, was 676, which is positive. This told us that our equation has two real solutions. Using the quadratic formula, we calculated these solutions to be x equals 2 and x equals negative 16 over 5. We verified both solutions by substituting them back into the original equation, confirming that they both satisfy the equation.