Problem 63 - Elementary and Intermediate Algebra

Question

Factor. If an expression is prime, so indicate. $$ 4 t^{2}-16 t+7 $$

Step-by-Step Solution

Answer

Factorized form: \(4(t - \frac{7}{2})(t - \frac{1}{2})\).

Step 1: Identify coefficients

The given quadratic expression is \(4t^2 - 16t + 7\). Identify the coefficients: \(a = 4\), \(b = -16\), and \(c = 7\).

Step 2: Calculate the discriminant

The discriminant \(D\) of a quadratic expression \(ax^2 + bx + c\) is given by \(b^2 - 4ac\). Calculate the discriminant: \((-16)^2 - 4(4)(7) = 256 - 112 = 144\).

Step 3: Check the discriminant

If the discriminant \(D\) is a perfect square, the quadratic can be factored over the set of real numbers. Here, \(144\) is a perfect square since \(144 = 12^2\).

Step 4: Use the quadratic formula to find roots

To factor, find the roots using the quadratic formula \(t = \frac{-b \pm \sqrt{D}}{2a}\). Substituting values, \(t = \frac{16 \pm \sqrt{144}}{8} = \frac{16 \pm 12}{8}\). This gives roots \(t = \frac{28}{8} = \frac{7}{2}\) and \(t = \frac{4}{8} = \frac{1}{2}\).

Step 5: Write the factorization using roots

The expression can thus be factored using its roots: \(4t^2 - 16t + 7 = 4(t - \frac{7}{2})(t - \frac{1}{2})\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool when dealing with quadratic equations in the form \(ax^2 + bx + c = 0\). It allows us to find the roots of the equation without needing to factor it manually. The formula is as follows: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation. The symbol \(\pm\) means that there will generally be two solutions: one with addition and one with subtraction.

• Use this formula when factoring is complex or when you need an exact solution.
• The formula works for any quadratic equation, as long as the discriminant (under the square root) is not negative.

Applying the quadratic formula helps in quickly identifying the roots of the quadratic equation, which is essential for factoring.

Discriminant

The discriminant is part of the quadratic formula and is crucial in determining the nature of the roots of a quadratic equation. The discriminant \(D\) is calculated as: \[ D = b^2 - 4ac \] It gives essential information about the roots:

• When \(D > 0\), there are two distinct real roots.
• When \(D = 0\), there is one real root (a repeated root).
• When \(D < 0\), there are two complex roots (no real roots).

In our original problem, the discriminant was calculated as \(144\), which is positive and also a perfect square. This means the quadratic has two real and rational roots, making it possible to factor it over the real numbers.

Roots of Quadratic Equations

Finding the roots of a quadratic equation is often the goal when solving these expressions. Roots are the values of \(t\) that make the equation true when plugged in. In the example provided, after applying the quadratic formula, we found the roots \(t = \frac{7}{2}\) and \(t = \frac{1}{2}\). To factor the quadratic equation using these roots, we express the equation in the form of \(a(t - r_1)(t - r_2)\), where \(r_1\) and \(r_2\) are the roots.

• Start by finding the roots using the quadratic formula, if not easily apparent through factoring.
• Express the quadratic equation as a product of binomials using these roots.
• This step often involves multiplying through by the leading coefficient \(a\) to ensure the expression is equivalent to its original form.

In our factorization, it resulted in \(4(t - \frac{7}{2})(t - \frac{1}{2})\), effectively expressing the original quadratic equation using its roots.