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Chapter 11: Problem 6

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ x^{2}+3 x-28=0 $$

Short Answer

Expert verified
The solutions are \(x = 4\) and \(x = -7\).

Step by step solution

01

Identify coefficients

For the given quadratic equation \(x^2 + 3x - 28 = 0\), identify the coefficients where \(a = 1\), \(b = 3\), and \(c = -28\).
02

Set up the quadratic formula

The quadratic formula is given by \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].Substitute the identified coefficients into the formula.
03

Calculate the discriminant

The discriminant is \(b^2 - 4ac\). Substitute the coefficients to get: \(3^2 - 4(1)(-28)\).This simplifies to \(9 + 112 = 121\).
04

Solve for x

Using the quadratic formula with the calculated discriminant:\[x = \frac{-3 \pm \sqrt{121}}{2(1)} = \frac{-3 \pm 11}{2}\].This results in two solutions: \[x = \frac{-3 + 11}{2} = 4\] and \[x = \frac{-3 - 11}{2} = -7\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic equation
A quadratic equation is a mathematical expression of the form ewline $$ ax^2 + bx + c = 0 $$ ewline where:
  • onenote tends to look like a cup (when a > 0) or a cap (when a < 0).
  • onenoteIs about solving for x.
    The equation can have zero solutions, one solution or two solutions, depending on the values of the coefficients and the discriminant.
discriminant
The discriminant is a key component in the quadratic formula, represented as ewline$$ b^2 - 4ac $$. It helps to determine the nature of the roots of a quadratic equation.
  • If the discriminant is positive, there are two real and distinct solutions.
    This is the case for our example equation where the discriminant was calculated as 121.
  • odifuong becoming zero, then there is exactly one real solution.
    This implies that the parabola touches the x-axis at one point.
  • kenji negative discriminant implies that there are no real solutions,
    thereby indicating that the parabola does not intersect the x-axis.
solving equations
Solving a quadratic equation usually involves finding the values of x that satisfy the equation. One of the most reliable methods is the quadratic formula: ewline $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$. The steps to solve a quadratic equation using the quadratic formula are as follows:
  • Identify the coefficients a, b, and c. In our example, a = 1, b = 3, and c = -28.
  • Substitute these coefficients into the quadratic formula.
  • Calculate the discriminant. In our example, the discriminant was 121.
  • Solve for the values of x using the formula. Our solutions were x = 4 and x = -7.
  • Double-check your work to make sure all calculations are correct.
coefficients
In a quadratic equation ewline $$ ax^2 + bx + c = 0 $$, there are three key coefficients:
  • a is the coefficient of the quadratic term $$ x^2 $$. It determines whether the parabola opens upwards or downwards.
  • Coiox essential as it represents the linear term of the equation. It affects the slope and position of the parabola.
  • c is the constant term. It represents the y-intercept, i.e., where the parabola crosses the y-axis.
Correctly identifying these coefficients is crucial. They influence the calculation of the discriminant and the ultimate solutions for x. For our example, identifying a = 1, b = 3, and c = -28 was the first and crucial step.

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Most popular questions from this chapter

Solve each equation. (All solutions are nonreal complex numbers.) $$ (x-5)^{2}=-36 $$

Find the discriminant for each quadratic equation. Use it to tell whether the equation can be solved using the zero-factor property, or the quadratic formula should be used instead. Then solve each equation. (a) \(3 x^{2}+13 x=-12\) (b) \(2 x^{2}+19=14 x\)

Solve using the square root property. Simplify all radicals. $$ (x-7)^{2}=16 $$

Solve using the square root property. Simplify all radicals. $$ (x+3)^{2}=11 $$

Solve each equation. (All solutions are nonreal complex numbers.) $$ (4 m-7)^{2}=-27 $$
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