Problem 48 - Algebra A Combined Function

Question

Solve each equation. $$ x^{2}-26=-11 x $$

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -13\).

Step 1: Move all terms to one side of the equation

Start by moving all terms to the left side of the equation so that one side equals zero. This helps in factoring or using the quadratic formula afterwards.Original equation: \[ x^2 - 26 = -11x \]Add \(11x\) to both sides:\[ x^2 + 11x - 26 = 0 \]

Step 2: Identify the coefficients

In the quadratic equation genre \( ax^2 + bx + c = 0 \), identify the values for \(a\), \(b\), and \(c\).Here, \(a = 1\), \(b = 11\), \(c = -26\).

Step 3: Use the quadratic formula

Since the equation might not factor easily, use the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute the values:\[ x = \frac{-11 \pm \sqrt{11^2 - 4(1)(-26)}}{2(1)} \]

Step 4: Calculate the discriminant

Calculate the discriminant \(b^2 - 4ac\):\[ 11^2 - 4(1)(-26) = 121 + 104 = 225 \]

Step 5: Solve the quadratic formula

Now substitute the value of the discriminant back into the quadratic formula:\[ x = \frac{-11 \pm \sqrt{225}}{2} \]Since \(\sqrt{225} = 15\), the formula becomes:\[ x = \frac{-11 \pm 15}{2} \]This results in two potential solutions:1. \( x = \frac{-11 + 15}{2} = 2 \)2. \( x = \frac{-11 - 15}{2} = -13 \)

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations, especially when they cannot be factored easily. A quadratic equation is typically set in the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). The quadratic formula is written as:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula provides us with the solutions (roots) of the quadratic equation. The expression under the square root, \( b^2 - 4ac \), is known as the discriminant, which determines the nature of the roots.

• When the discriminant is positive (\( b^2 - 4ac > 0 \)), there are two distinct real roots.
• If the discriminant is zero (\( b^2 - 4ac = 0 \)), there is exactly one real root, usually referred to as a repeated or double root.
• Finally, if the discriminant is negative (\( b^2 - 4ac < 0 \)), the equation has no real roots, but two complex roots.

Make use of the quadratic formula when you can't factor the quadratic equation easily or when you want an exact solution.

Discriminant

The discriminant is a significant component of the quadratic formula and gives vital information about the nature of the roots of a quadratic equation. It is expressed as:
\[b^2 - 4ac\]
For the quadratic equation \( ax^2 + bx + c = 0 \), calculating the discriminant lets you foresee the types of solutions you will obtain without fully solving the equation.

• If \( b^2 - 4ac > 0 \), you'll find two distinct real roots. This implies the parabola represented by the quadratic function intersects the x-axis at two points.
• If \( b^2 - 4ac = 0 \), there is one real root, indicating the parabola just "touches" the x-axis, creating a perfect square trinomial.
• If \( b^2 - 4ac < 0 \), no real roots exist, but rather, two complex solutions. This means the parabola does not intersect the x-axis at all.

Understanding the discriminant helps in quickly assessing whether you’re dealing with real or complex solutions and how many solutions you should expect.

Factoring Quadratic Equations

Factoring is a method used to solve quadratic equations by converting the quadratic equation into a product of simpler expressions, typically when the equation can be rearranged in the form \( ax^2 + bx + c = 0 \). Not all quadratic equations can be easily factored, so recognizing when to use this method is essential.
The main steps in factoring involve:
- Expressing the equation as a product of two binomials: \((px + q)(rx + s) = 0\).- Setting each binomial equal to zero gives the solutions, as one or both must equal zero for the product to be zero.

• Start by checking for a Greatest Common Factor (GCF), simplifying the equation if possible.
• Identify pairings of numbers that multiply to give you the product \( ac \) but add up to \( b \). This is key in factorable quadratics.
• Rearrange the middle term using the pair found above and factor by grouping.

Factoring is most effective in cases where coefficients are straightforward or easily reducible, and solutions can often be found quicker than using the quadratic formula.