Problem 79 - Algebra A Combined Function

Question

Write a quadratic equation in standard form that has two solutions, 5 and 7 .

Step-by-Step Solution

Answer

The quadratic equation is \( x^2 - 12x + 35 = 0 \).

Step 1: Understanding the Roots

Given roots of the quadratic equation are 5 and 7. This means if one solution of the equation is 5 and another is 7, then the equation can be expressed as \[ (x - 5)(x - 7) = 0 \] which shows the factors of the equation that give these roots.

Step 2: Expand the Factors

Expand the factors to write the equation in the form of \( ax^2 + bx + c = 0 \).To expand \( (x - 5)(x - 7) \): - Multiply the first terms: \( x \times x = x^2 \)- Multiply the outer terms: \( x \times (-7) = -7x \)- Multiply the inner terms: \( -5 \times x = -5x \)- Multiply the last terms: \( -5 \times (-7) = 35 \)Combine them to get the expanded form: \[ x^2 - 7x - 5x + 35 = 0 \]

Step 3: Simplify the Expression

Combine like terms from the expanded expression. The expression \( x^2 - 7x - 5x + 35 \) simplifies to:\[ x^2 - 12x + 35 = 0 \]This is the quadratic equation in its standard form, with coefficients a = 1, b = -12, and c = 35.

Key Concepts

Standard Form of a Quadratic Equation

A quadratic equation is a polynomial equation of degree 2. The standard form of a quadratic equation is: \[ ax^2 + bx + c = 0 \]where:

• \( a \), \( b \), and \( c \) are constants,
• \( x \) represents the variable or unknown,
• \( a \) cannot be zero.

This form is very important since it allows us to apply a variety of methods to find the roots of the equation, such as factoring, using the quadratic formula, or completing the square. When you convert a quadratic equation into standard form, you can immediately identify the coefficients that you will need for these methods.

Expanding Factors in Quadratic Equations

To convert a factored form of a quadratic equation into standard form, we need to expand the factors. Expansion involves multiplying the terms together, which transforms the product of binomials into a polynomial. Let's look at how we expand the given factors:For the factors \((x - 5)(x - 7)\), follow these steps:

• Multiply the first terms: \( x \times x = x^2 \)
• Multiply the outer terms: \( x \times (-7) = -7x \)
• Multiply the inner terms: \( -5 \times x = -5x \)
• Multiply the last terms: \( -5 \times (-7) = 35 \)

Once you have all these products, combine them. You will notice there are like terms, which are \(-7x\) and \(-5x\) in this case. Combine to get \(-12x\), leading to the quadratic expression:\[ x^2 - 12x + 35 = 0 \]

Quadratic Solutions

Solutions of a quadratic equation are the values of \( x \) that satisfy the equation. In the factored form, solutions are more apparent. For example, considering our equation\[(x - 5)(x - 7) = 0\],the solutions or roots are the values that make each factor equal to zero:

• If \( x - 5 = 0 \), then \( x = 5 \)
• If \( x - 7 = 0 \), then \( x = 7 \)

These solutions are the x-values where the curve of the quadratic equation crosses the x-axis. Factored equations conveniently reveal the roots directly. However, in standard form, we might need to use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) to find the solutions. Quadratic solutions can be real or complex, and they give us valuable insights into the nature of the quadratic function and its graph.