Question

What are the two solutions for \(x\) in the equation \(4 x^{2}+8 x-4=0\) (A) \(x=3 \sqrt{2}\) and \(x=-4\) (B) \(x=2 \sqrt{3}\) and \(x=\sqrt{11}+13\) (C) \(x=-\sqrt{7}\) and \(x=-5\) (D) \(x=-1-\sqrt{2}\) and \(x=\sqrt{2}-1\)

Short answer

The solutions are (D) \( x = \sqrt{2} - 1 \) and \( x = -1 - \sqrt{2} \).

Step-by-step solution

Identify the Quadratic Formula

The given quadratic equation is in the form \( ax^2 + bx + c = 0 \), where \( a = 4 \), \( b = 8 \), and \( c = -4 \). To solve for \( x \), we use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Compute the Discriminant

Calculate the discriminant, \( b^2 - 4ac \), to determine the nature of the roots. Substitute \( a = 4 \), \( b = 8 \), and \( c = -4 \): \[ b^2 - 4ac = 8^2 - 4 \times 4 \times (-4) = 64 + 64 = 128 \].

Apply the Quadratic Formula

Substitute the values into the quadratic formula: \[ x = \frac{-8 \pm \sqrt{128}}{8} \]. Simplify \( \sqrt{128} \) to \( 8\sqrt{2} \), so the equation becomes: \[ x = \frac{-8 \pm 8\sqrt{2}}{8} \].

Simplify the Expression

Simplify \( \frac{-8 \pm 8\sqrt{2}}{8} \) to get the solutions for \( x \). Divide each term separately: \[ x = \frac{-8}{8} \pm \frac{8\sqrt{2}}{8} = -1 \pm \sqrt{2} \].

Interpret the Solutions

The solutions are \( x = -1 + \sqrt{2} \) and \( x = -1 - \sqrt{2} \). These correspond to option (D): \( x = \sqrt{2} - 1 \) and \( x = -1 - \sqrt{2} \).

Key concepts

Understanding the Quadratic Formula

The quadratic formula is a powerful tool to find solutions for quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This handy formula enables you to solve for \(x\) when the equation does not factor easily. The formula is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Where:

• \(a\), \(b\), and \(c\) are coefficients from the equation.
• \(b^2 - 4ac\) is called the discriminant.

The formula delivers two possible solutions because of the "\(\pm\)" sign within it, meaning you calculate one solution using the positive sign and another with the negative. By mastering this formula, you can efficiently tackle any quadratic equation you encounter.

Exploring the Discriminant

The discriminant, symbolized as \(b^2 - 4ac\), plays a pivotal role in using the quadratic formula. It is the part under the square root in the formula and tells you the nature of the roots of the quadratic equation. Computing the discriminant allows us to understand what kind of solutions to expect without even solving the equation. Here are the possible scenarios:

• If \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), the equation has exactly one real root, often called a repeated or double root.
• If \(b^2 - 4ac < 0\), the equation does not have real roots but instead has two complex roots.

In our exercise example, the discriminant was 128, indicating two distinct real roots.

Nature of Roots Explained

The nature of the roots tells us about the solutions of quadratic equations in terms of type and number. Understanding this helps in predicting the roots before calculation. Here is how the discriminant determines this nature: - **Two Distinct Real Roots:** When \(b^2 - 4ac > 0\). The square root part yields a positive number, allowing us to have two separate solutions after applying the quadratic formula.- **One Real Double Root:** When \(b^2 - 4ac = 0\). The square root part becomes zero, and both solutions from the quadratic formula will be identical, giving us a single root.- **Complex Roots:** When \(b^2 - 4ac < 0\). Since you cannot square root a negative, the roots involve imaginary numbers, leading to complex solutions. For our equation, the discriminant was positive, confirming two separate real solutions for \(x\).

Simplifying Expressions

Simplifying expressions is the process of making an expression easier to understand and work with. In the context of solving quadratic equations using the quadratic formula, this often involves managing square roots and fractions. Here is how it's done: - First, simplify the square root if possible. For instance, \(\sqrt{128}\) is simplified to \(8\sqrt{2}\).- Next, carefully perform any arithmetic with fractions. Break down components to ensure correct division, like dividing \(-8\) and \(8\sqrt{2}\) by 8, leading to \(-1\) and \(\sqrt{2}\). By simplifying expressions, solutions are made neat and clear for easier interpretation. The final solutions from our example become \(x = -1 + \sqrt{2}\) and \(x = -1 - \sqrt{2}\), after complete simplification.