Question

Use the Quadratic Formula to solve the quadratic equation. $$ 16 x^{2}+8 x-3=0 $$

Short answer

The solutions to the equation \(16x^{2}+8x-3=0\) are \(x = -0.25\) and \(x = 0.75\).

Step-by-step solution

Identify the coefficients

In the quadratic equation \(16x^{2}+8x-3=0\), the coefficients are \(a = 16\), \(b = 8\), and \(c = -3\).

Plug the coefficients into the Quadratic Formula

The equation for the Quadratic Formula is \(x = \frac{-b ± \sqrt{b^{2} - 4ac}}{2a}\). Substitute the coefficients from step 1 and calculate the square root part, or the discriminant: \( \sqrt{b^{2} - 4ac} = \sqrt{8^{2} - 4*16*(-3)} = \sqrt{64 + 192} = \sqrt{256}\).

Calculate the values of x

Substitute \(b = 8\) and \( \sqrt{256} = 16 \) into the Quadratic Formula to find two solutions. \(x = \frac{-8 ± 16}{2*16} = -0.25, 0.75.\)

Key concepts

Quadratic Equation

A quadratic equation is a fundamental concept in algebra, typically presented in the standard form: \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The "quadratic" term stems from the "square" of \(x\) in \(ax^2\). These equations represent parabolas when graphed, which can open upwards or downwards depending on the sign of \(a\).
Quadratic equations often arise in various real-life scenarios, including physics, engineering, and economics, whenever a relationship involving squares is analyzed.
Solving quadratic equations can be achieved using several methods, such as factoring, completing the square, or the quadratic formula. Each method has its specific applications, but the quadratic formula is a universal technique that works for any quadratic equation. Because it handles cases both with rational and irrational roots efficiently, it's a go-to method for solving quadratics when factoring is cumbersome or infeasible.

Discriminant

The discriminant is a key part of the quadratic formula and plays a crucial role in understanding the nature of the roots of a quadratic equation. It is found under the square root in the quadratic formula:
\[ \Delta = b^2 - 4ac \].
The discriminant tells us whether the solutions to the quadratic equation are real or complex, and whether they are distinct or repeated.

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root, also known as a repeated root.
• If \(\Delta < 0\), the equation has two complex conjugate roots.

Analyzing the discriminant before solving a quadratic equation can give us valuable insight into what type of solutions to anticipate, saving time and effort in computations. Knowing this, we can decide the best approach to tackle the equation effectively.

Coefficient Identification

Coefficient identification is a vital first step in solving quadratic equations using the quadratic formula. In the equation \(ax^2 + bx + c = 0\), the numbers \(a\), \(b\), and \(c\) are called coefficients.
Identifying these correctly is crucial because they directly dictate the equation's behavior and solutions.

• \(a\) is the coefficient of \(x^2\), often called the leading coefficient, which determines the direction of the parabola (upward for positive, downward for negative).
• \(b\) is the coefficient of \(x\), influencing the symmetry of the parabola and its axis of symmetry.
• \(c\) is the constant term, which affects the vertical translation of the parabola on a graph.

In our given equation \(16x^{2}+8x-3=0\), identifying these coefficients as \(a = 16\), \(b = 8\), and \(c = -3\) correctly provides the foundation for applying the quadratic formula accurately. Incorrect identification may lead to errors in computation and incorrect solutions.