Problem 52 - Algebra 1: Concepts and Skills

Question

Solve the quadratic equation. $$ x^{2}-6 x-10=0 $$

Step-by-Step Solution

Answer

The roots of the equation \(x^{2}-6 x-10=0\) are \(x = 3 + \sqrt{19}\) and \(x = 3 - \sqrt{19}\).

Step 1: Identifying the coefficients

We compare the equation with the standard form of a quadratic equation \(ax^2 + bx + c = 0\) and identify the coefficients. Here, \(a = 1\), \(b = -6\) and \(c = -10\).

Step 2: Substitute in Quadratic formula

Next, we substitute these values into the quadratic formula to solve for \(x\). The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Replace \(a\), \(b\) and \(c\) in the formula with 1, -6 and -10 Respectively.

Step 3: Evaluate

After substituting the values, we get \(x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot -10}}{2 \cdot 1}\). This simplifies to \(x = \frac{6 \pm \sqrt{36 + 40}}{2}\) and further into \(x = \frac{6 \pm \sqrt{76}}{2}\).

Step 4: Factor out

To Simplify it further, we factor out \( \frac{2}{2}\) from entire equation. This results into \(x = 3 \pm \sqrt{19}\).

Key Concepts

Quadratic Formula

Solving quadratic equations can often be achieved using the quadratic formula. This method is especially valuable when the equation cannot easily be factored. The quadratic equation is given in the form:

• \( ax^2 + bx + c = 0 \)

Where \(a\), \(b\), and \(c\) are coefficients. To find the values of \(x\) that satisfy the equation, we use the quadratic formula, which is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's what each part of the quadratic formula does:

• \(-b\) starts the calculation by changing the sign of the coefficient \(b\).
• \(\pm\) signifies that you will calculate two possible solutions, one using addition and one using subtraction.
• \(\sqrt{b^2 - 4ac}\) involves taking the square root of the discriminant \((b^2 - 4ac)\). If the discriminant is positive, there are two real solutions.
• The whole expression is divided by \(2a\), which scales the solutions appropriately.

Utilizing the quadratic formula provides a systematic solution approach and works with all quadratic equations, even when factoring is challenging.

Standard Form of a Quadratic Equation

Understanding the standard form of a quadratic equation is crucial for solving these types of problems. Quadratic equations may appear in different forms, but the standard form is always written as:

• \( ax^2 + bx + c = 0 \)

In this notation:

• \(a\) is the coefficient of \(x^2\) and must not be zero since the equation wouldn't remain quadratic.
• \(b\) is the coefficient of \(x\), which modifies the linear part of the equation.
• \(c\) is the constant term, which shifts the equation graph vertically.

Identifying \(a\), \(b\), and \(c\) correctly simplifies the process of utilizing methods like the quadratic formula or factoring to find the equation's solutions. Being adept at recognizing the standard form can aid you in efficiently approaching and solving various quadratic problems.

Solving by Factoring

Factoring is a common technique used to solve quadratic equations when the equation can be rewritten as a product of two binomials. Here’s how to factor when solving a quadratic equation:Firstly, rearrange the quadratic equation in the standard form:

• \( ax^2 + bx + c = 0 \)

Next, look for two numbers that multiply to the constant term \(c\) and add up to the linear coefficient \(b\). For some problems, like the one in the exercise, factoring might not be straightforward. When factoring is possible, rewrite the equation as:

• \((px + q)(rx + s) = 0\)

Where \(p\), \(q\), \(r\), and \(s\) are numbers you find by inspection. Each factor can subsequently be set to zero:

• \(px + q = 0\)
• \(rx + s = 0\)

Solving these equations gives the solutions to the quadratic equation. Although factoring is highly effective, not all quadratics are factorable. In such cases, the quadratic formula is used instead. Being comfortable with both methods gives you flexibility in solving different types of quadratic equations.