Problem 69 - Algebra 1: Concepts and Skills

Question

Choose a method and solve the quadratic equation. Explain your choice. $$ 2 x^{2}+7 x+3=0 $$

Step-by-Step Solution

Answer

The solutions for the equation \(2x^{2}+7x+3=0\) are \(x= -0.5\) and \(x= -3\).

Step 1: Identify the coefficients

The coefficients for this quadratic equation are \(a = 2\), \(b = 7\), and \(c = 3\). These values will be substituted in the quadratic formula to solve for \(x\).

Step 2: Substitute the coefficients into the Quadratic formula

Now, substitute \(a = 2\), \(b = 7\), and \(c = 3\) into the quadratic formula: \[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\] which gives: \[x = \frac{-7 \pm \sqrt{7^{2}-4*2*3}}{2*2}\]

Step 3: Solving the formula

Next, perform the operations \(7^{2}-4*2*3 = 49-24 = 25\) and \(2*2 = 4\), giving: \[x = \frac{-7 \pm \sqrt{25}}{4}\] Simplify further by taking the square root of 25, which gives: \[x = \frac{-7 + 5}{4}\] and \[x = \frac{-7 - 5}{4}\] Finally, this simplifies to \(x = -0.5\) and \(x = -3\) respectively.

Key Concepts

Quadratic Formula

The quadratic formula is a robust tool for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). This formula helps find the equation's roots by using the coefficients \(a\), \(b\), and \(c\). The quadratic formula is expressed as:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula works universally for quadratics, regardless of whether they are factorable. All you need is to substitute the values of \(a\), \(b\), and \(c\) into the formula. This method is especially useful when a quadratic doesn't factor nicely using integers. Often students use it when other methods become cumbersome. Just remember, if the expression under the square root (the discriminant \(b^2 - 4ac\)) is negative, the roots will be complex, not real numbers.

Solving Quadratics

Solving quadratics involves finding the values of \(x\) that make the equation true, where the equation is typically expressed as \(ax^2 + bx + c = 0\). There are various methods to solve quadratics including:

• Factoring: This is attempted first if the quadratic is easily factorable.
• Quadratic Formula: A universal method for any quadratic equation.
• Completing the Square: Useful for equations that are difficult to factor.

Each method has its pros and cons, and the choice may depend on the specific equation. The goal is to rewrite the equation to a form where you can easily see the solutions or roots of \(x\). A solved quadratic will typically have two roots, which could be real or complex.

Factorization

Factorization involves expressing the quadratic equation \(ax^2 + bx + c = 0\) as a product of two binomials. This method works well when the quadratic can be broken down into easier-to-handle expressions like \((x + p)(x + q) = 0\). Solving each binomial for zero will give the values of \(x\) where:

• \(p\) and \(q\) are integers that multiply to \(ac\).
• The sum of \(p\) and \(q\) is \(b\).

This method is particularly effective when it becomes apparent that \(a\), \(b\), and \(c\) allow easy factoring. When you can spot a pattern or quickly guess the factors of \(ac\) that add up to \(b\), it's a huge time saver! However, not all quadratic equations are factorable using integers, so other methods may be necessary.

Roots of Equations

The roots of a quadratic equation are the values of \(x\) that make the equation true. In other words, they are the points where the quadratic function crosses the x-axis. For the quadratic equation \(ax^2 + bx + c = 0\), the roots can be real or complex numbers depending on the discriminant \(b^2 - 4ac\).
If the discriminant is:

• Positive: There are two distinct real roots.
• Zero: There is exactly one real root (the roots are repeated).
• Negative: The roots are complex or imaginary.

Understanding and calculating these roots is pivotal in solving the quadratic equation. It helps in various applications such as physics, engineering, and several fields requiring trajectory or optimization analysis.