Question

The equation \(\frac{1}{4}\left(4 x^{2}-8 x-k\right)\) has two solutions: \(x=-5\) and \(x=7 .\) What is the value of \(2 k ?\)

Short answer

The value of \(2k\) is 280.

Step-by-step solution

Identify the factors

The given equation has solutions at \(x = -5\) and \(x = 7\). Hence, the quadratic can be written as \((x + 5)(x - 7)\).

Expand the factors

Expand the expression \((x + 5)(x - 7)\) to get the quadratic form: \(x^2 - 7x + 5x - 35 = x^2 - 2x - 35\).

Relate to the given quadratic expression

Equate the standard form \(4 x^{2}-8 x-k\) with the expanded form after considering the coefficient and constant terms: \(4(x^2 - 2x - 35)\).

Distribute the coefficient

Distribute the coefficient of 4: \(4(x^2 - 2x - 35) = 4x^2 - 8x - 140\).

Compare constants to find k

Compare \(4x^2 - 8x - k\) with \(4x^2 - 8x - 140\). Hence, \(k = 140\).

Calculate 2k

Double the value of \(k\): \(2k = 2 \times 140 = 280\).

Key concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form \(ax^2 + bx + c = 0\). The highest exponent is 2, making the equation quadratic. In the given exercise, the equation can be represented as \(4x^2 - 8x - k\). Quadratic equations often describe parabolic graphs. Understanding how to manipulate and solve these equations is key, especially for SAT math prep.
Quadratic equations can be solved through multiple methods, including:

• Factoring
• Using the Quadratic Formula
• Completing the Square
• Graphing

Depending on the problem, one method may be easier or more efficient than others.

Factoring

Factoring makes solving quadratic equations more straightforward. Essentially, you are expressing the quadratic equation as a product of its linear factors, such as \( (x + m)(x + n) = 0 \). In our exercise, the given solutions are \(x = -5\) and \(x = 7\). Therefore, the quadratic equation can be expressed as \( (x + 5)(x - 7) \).
Steps to factor a quadratic equation:

• Identify the solutions or roots.
• Express the quadratic equation as a product of linear factors using the roots.
• Simplify the expression if required.
• Check if the factorization is correct by expanding it back to its standard form.

Factoring is useful when the quadratic equation can be easily decomposed into simpler binomials.

Algebraic Solutions

Once the quadratic equation is factored, finding the solutions involves simple algebraic manipulation. These solutions are values of \(x\) that make the equation true. To solve the exercise provided, we:

• Translated the given solutions \(x = -5\) and \(x = 7\) into linear factors \( (x + 5)(x - 7) \).
• Expanded the factors to get the quadratic form: \( x^2 - 2x - 35 \).
• Matched it with the given form \( \frac{1}{4}(4x^2 - 8x - k) \).
• Distributed the coefficient of 4 to both sides to form \( 4x^2 - 8x - 140 \).

With the equations matched, correct algebraic manipulation revealed that \(k = 140\). Doubling it, we get \(2k = 280\). This process ensures understanding of how changes in coefficients and constants will impact the equation.

SAT Math

SAT math covers a wide array of topics, including quadratic equations. Mastery of factoring, expanding, and solving quadratic equations is essential because these principles frequently appear in SAT questions. To ace SAT math:

• Familiarize yourself with common algebraic forms and concepts like factoring.
• Practice identifying roots of quadratic equations quickly.
• Get comfortable with algebraic manipulations and distributions of coefficients as shown in the exercise.
• Use timed practice tests to improve speed and accuracy.

Whether it’s factoring or graphing quadratic equations, proficiency in these areas boosts your SAT score and enhances your overall math competency.