Problem 67 - Algebra 1

Question

Use the quadratic formula to solve the equation. $$2 x^{2}+4 x-3=0$$

Step-by-Step Solution

Answer

The roots of the equation are \(x = -1 + \sqrt{10}\) and \(x = -1 - \sqrt{10}\).

Step 1: Identify the coefficients

The equation \(2x^2 + 4x - 3 = 0\) is a quadratic equation. The coefficients are \(a = 2\), \(b = 4\), and \(c = -3\).

Step 2: Compute the discriminant

For the quadratic formula, we need to calculate the value of the discriminant, which is \(b^2 - 4ac\). By substituting our values into the equation, we get \(16 - 4*2*(-3) = 16 + 24 = 40\)

Step 3: Find the roots using the quadratic formula

The roots of the equation can be found using the quadratic formula \(-b \pm \sqrt{b^2 - 4ac} \over 2a\). Substituting the earlier computed values into this formula, we get the roots of the equation as \(x = -4 \pm \sqrt{40} \over 4\). This simplifies to \(x = -1 \pm \sqrt{10}\)

Key Concepts

quadratic equations

Quadratic equations are polynomial equations of the second degree. Usually, they appear in the form \(ax^2 + bx + c = 0\). Here, the constants \(a\), \(b\), and \(c\) are coefficients. Importantly, \(a\) should not be zero, since the equation would then become linear rather than quadratic. Quadratic equations are fundamental in algebra and appear frequently in various fields such as physics, engineering, and finance.

The expression \(x^2\) in the equation is what makes it a quadratic equation. Solving these types of equations involves finding the values of \(x\) that make the equation true. These values are known as the "roots" or "solutions" of the equation, and there are typically two solutions in real numbers.

discriminant

In the context of quadratic equations, the discriminant is a key component that helps determine the number and type of solutions (or roots). It is calculated using the formula \(b^2 - 4ac\).

The value of the discriminant can tell us a lot about the roots:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is exactly one real root (also known as a repeated or "double" root).
• If the discriminant is negative, there are no real roots; instead, there are two complex roots.

For the given example, the discriminant is 40, which is positive, indicating that the equation \(2x^2 + 4x - 3 = 0\) has two distinct real roots.

solving quadratic equations

Solving quadratic equations can be achieved by applying several techniques; one of the most precise is using the quadratic formula. The formula is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here, \(-b\) indicates a negation of the linear coefficient, and \(\pm\) accounts for the equation's two possible solutions. Using the quadratic formula provides a straightforward way to find the roots, regardless of whether they are real or complex.

In the problem we are solving, we calculated that the discriminant is 40. Plugging the values of \(a\), \(b\), and \(c\) into the quadratic formula gives us the solutions. When simplifying these plug-ins, we get \(x = -1 \pm \sqrt{10}\), offering two solutions: \(-1 + \sqrt{10}\) and \(-1 - \sqrt{10}\). This represents the points at which the quadratic equation equals zero. The quadratic formula is reliable, providing exact answers to any quadratic problem, given the necessary computations are done correctly.