Problem 45 - Algebra 1: Concepts and Skills

Question

Solve the quadratic equation. $$ x^{2}+10 x-3=0 $$

Step-by-Step Solution

Answer

The solutions of the quadratic equation are \(x1 = 0.29\) and \(x2 = -10.29\)

Step 1: Identify coefficients

First, identify the coefficients a, b, and c from the given quadratic equation. In this case, a=1, b=10, and c=-3.

Step 2: Substitute coefficients into the quadratic formula

Next, substitute a, b, and c into the quadratic formula. This gives \(x = [-10 ± sqrt{(10^2 - 4*1*(-3))}]/(2*1)\)

Step 3: Solve the equation

Simplify this to calculate the solution for x. Performing the operation in the square root gives us: \(x = [-10 ± sqrt{(100 + 12)}]/2\), from which \(x = [-10 ± sqrt{112}]/2\). This simplifies to \(x = [-10 ± 10.58]/2\), yielding two solutions: \(x1 = 0.29\) and \(x2 = -10.29\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool that allows us to find the solutions to any quadratic equation of the form \( ax^2 + bx + c = 0 \). It is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides the solutions, or the "roots," of the quadratic equation by plugging in the values of the coefficients \(a\), \(b\), and \(c\). These coefficients represent the numbers in front of the quadratic, linear, and constant terms of the equation respectively.The term \( \pm \) means that there are usually two possible solutions, corresponding to the two different signs. The expression inside the square root, \(b^2 - 4ac\), is known as the "discriminant," and it plays a crucial role in determining the nature of the roots. A positive discriminant indicates real and distinct roots, zero means real and identical roots, and a negative value implies complex roots.

Coefficients

In a quadratic equation, coefficients are the numbers that multiply each term. They are crucial components that determine the shape and position of a parabola, which is the graphical representation of a quadratic equation. For the equation \(ax^2 + bx + c = 0\), each respective coefficient has a role:

• \(a\) is the coefficient of the quadratic term \(x^2\). If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.
• \(b\) is the coefficient of the linear term \(x\). It influences the slope or tilt of the parabola.
• \(c\) is the constant term, which represents the y-intercept of the parabola when \(x = 0\).

Identifying these coefficients is the first step in using the quadratic formula, as each one must be inserted precisely into the equation. In our example problem, we identified \(a = 1\), \(b = 10\), and \(c = -3\). Understanding these coefficients' roles helps in predicting the behavior of the quadratic equation before finding its roots.

Solving Equations

Like many algebraic tasks, solving a quadratic equation requires a systematic approach. Using the quadratic formula is one of the most widely applicable methods, especially when factoring is not feasible. By following the formula:1. Identify the coefficients \(a\), \(b\), and \(c\).2. Substitute into the quadratic formula.3. Compute the discriminant, \(b^2 - 4ac\).4. Solve for \(x\) by simplifying the values found in your calculations.Remember, the discriminant provides information about the nature of the solutions:

• If \(b^2 - 4ac > 0\), there are two distinct real solutions.
• If \(b^2 - 4ac = 0\), there is exactly one real solution (or a repeated root).
• If \(b^2 - 4ac < 0\), the solutions are complex or imaginary numbers.

In practical terms, solving for \(x\) means finding the values that make the equation true. These values can be graphed as the x-intercepts of the parabola.

Step-by-Step Solution

To reinforce understanding through a detailed example, let's solve the equation \(x^2 + 10x - 3 = 0\) step by step:1. **Identify coefficients**: First, pick out \(a = 1\), \(b = 10\), and \(c = -3\).2. **Substitute into the quadratic formula**: Plug these into the quadratic formula: \[ x = \frac{-10 \pm \sqrt{10^2 - 4 \times 1 \times (-3)}}{2 \times 1} \]3. **Simplify the discriminant**: Calculate under the square root:\[ b^2 - 4ac = 10^2 + 12 = 112 \]The positive discriminant predicts two real solutions.4. **Solve for \(x\)**:- Compute the combined operations: \[ x = \frac{-10 \pm \sqrt{112}}{2} \]- Simplify further to: \[ x = \frac{-10 \pm 10.58}{2} \]- Evaluate to find the solutions:\[ x_1 = 0.29 \quad \text{and} \quad x_2 = -10.29 \]These solutions correspond to the x-intercepts of the parabola. By following this structured approach, we have thoroughly solved the equation.