Question

Find the solution to the following quadratic equations using the quadratic formula.

A) f(x) = 5x^2 + 8x – 3 = 1
B) g(x) = -1/4 (x-4)(x+6)
C) h(x) = 2(x – 3)^2 - 32"

Answer 1

Quadratic equations are solved using the quadratic formula, which requires the equation to be in standard form ax² + bx + c = 0. The formulas and solutions change based on the coefficients a, b, and c of the specific equation.

Step-by-step explanation:

The Solution of Quadratic Equations

To solve quadratic equations, we use the quadratic formula which is:

x = (-b ± √(b² - 4ac)) / (2a)

To apply it, the equation must be in the form ax² + bx + c = 0. Let's solve the given equations:

A) f(x) = 5x² + 8x – 3 = 1

First, bring all terms to one side to get the standard form: 5x² + 8x – 4 = 0. Now, apply the quadratic formula with a = 5, b = 8, and c = -4 to find the roots of the equation.

B) g(x) = -1/4 (x-4)(x+6)

Expand the equation to get g(x) = -1/4x² + (3/2)x - (3/2). Apply the quadratic formula with a = -1/4, b = 3/2, and c = -3/2.

C) h(x) = 2(x – 3)² - 32

Expand the equation to get h(x) = 2x² - 12x + 18 - 32 which simplifies to 2x² - 12x - 14 = 0. Here, a = 2, b = -12, and c = -14. Apply the quadratic formula for the roots.

Remember to calculate discriminant b² - 4ac for each, to ensure there are real solutions, as these will determine the nature of the roots.

Answer 2

Using the quadratic formula, solutions to the quadratic equations can be found by first rewriting the given equations in the standard form ax²+bx+c = 0 and then substituting the corresponding coefficients into the formula.

Step-by-step explanation:

The quadratic formula is used to find solutions to quadratic equations of the form ax²+bx+c = 0. Let's solve the given quadratic equations:

• A) f(x) = 5x² + 8x – 3
  First, we set the equation equal to zero by subtracting 1 from both sides to get 5x² + 8x – 4 = 0
  Using the quadratic formula, x = –(b ± √(b²-4ac))/(2a), where a = 5, b = 8, and c = -4.

• B) g(x) = -1/4 (x-4)(x+6)
  First, we need to expand the equation: g(x) = -​1/4(x² + 2x - 24) = -​1/4x² - 1/2x + 6.
  Set to the standard form g(x) = 0, we have -​1/4x² - 1/2x + 6 = 0.

• C) h(x) = 2(x – 3)² - 32
  First, expand and simplify: h(x) = 2(x² - 6x+9) - 32 = 2x² - 12x + 18 - 32 = 2x² - 12x - 14.
  Set the equation to zero to get the standard form: 2x² - 12x - 14 = 0.

Once each quadratic equation is in the standard form, you can substitute values of a, b, and c into the quadratic formula to calculate the solutions.