Question

Solve the quadratic equation. $$4 x^{2}+16 x+15=0$$

Short answer

The solutions to the equation are \(x = -1.5\) and \(x = -2.5\).

Step-by-step solution

Identify coefficients

First identify the coefficients a, b, and c in the given equation. In this equation, \(4x^2+16x+15 = 0\), a = 4, b = 16, and c = 15.

Applying the Quadratic Formula

Next, substitute the values of a, b, and c into the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Using this formula givesx = \frac{-16 \pm \sqrt{16^2-4(4)(15)}}{2*4}= \frac{-16 \pm \sqrt{256-240}}{8}= \frac{-16 \pm \sqrt{16}}{8}.

Simplifying the result

Lastly, simplify this further to find the roots. This gives x1 = \frac{-16 + 4}{8} = -1.5 and x2 = \frac{-16 - 4}{8} = -2.5. This means the solutions of the equation are x = -1.5 and x = -2.5.

Key concepts

The Quadratic Formula

The quadratic formula is a fundamental tool in algebra for finding the roots of quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). When faced with an equation like \(4x^2 + 16x + 15 = 0\), the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) allows you to find the values of \(x\) that make the equation true.

To use it, you'll need to identify the coefficients – \(a\), \(b\), and \(c\) – which are the numerical factors before \(x^2\), \(x\), and the constant term, respectively. Here, \(a=4\), \(b=16\), and \(c=15\). After plugging these into the formula, simplifying is crucial. Simplifying the square root of the discriminant \(\sqrt{b^2-4ac}\) and the fractions can often reveal the roots in their simplest form.

Understanding and applying the quadratic formula is key to solving many algebraic problems, and while it may seem daunting at first, with practice it can become a straightforward and efficient method.

Factoring Quadratic Equations

Factoring is another method for solving quadratic equations, where you express the quadratic in the form \(ax^2 + bx + c\) as a product of binomials. For the equation \(4x^2 + 16x + 15 = 0\), one might look for two numbers that multiply to \(ac\) and add up to \(b\). However, in this case, factoring by inspection might be challenging due to the larger coefficients, so other methods such as the quadratic formula might be easier.

When factoring is straightforward, it's an efficient method as it can often be done mentally with practice and helps to understand the nature of the quadratic equation more deeply. Factoring can reveal the zeros of the function, or the x-intercepts of the graph, providing a clear visual representation of the solutions.

Coefficients of Quadratic Equations

In a quadratic equation, the coefficients are the numbers in front of the terms with variables. They are essential because they determine the shape and position of the parabola when the quadratic equation is graphed. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), with \(a\), \(b\), and \(c\) as the coefficients.

In our example \(4x^2 + 16x + 15 = 0\), \(4\) is the coefficient of \(x^2\), which affects the width and the direction of the parabola. A positive \(a\) means the parabola opens upward, while a negative \(a\) would open it downward. The coefficient \(16\) of \(x\) influences the position of the vertex along the x-axis, and \(15\) is the constant term that moves the parabola up or down the y-axis.

By understanding how these coefficients interact with the graph of the equation, one can predict the number and nature of the roots without even solving it. If \(a\) and \(c\) have the same sign, the quadratic will have no real roots. If they have opposite signs, there are two real roots, as in the example given.