Question

Solve the quadratic equation by factoring. $$ 16 x^{2}+56 x+49=0 $$

Short answer

The roots of the quadratic equation \(16x^{2}+56x+49=0\) are both \(x = -1.75\).

Step-by-step solution

Express the quadratic in standard format

A quadratic equation is usually written in the standard form \(ax^{2}+bx+c=0\). It's important to identify a, b, and c. For this equation, \(16x^{2} + 56x + 49 = 0\), a = 16, b = 56, and c = 49

Factor the quadratic equation

The quadratic equation takes the form \((4x+7)^{2}=0\). Since the product of two factors is zero, at least one of the factors must be zero.

Solve for x

Set the factor equal to zero and solve for \(x\), so we get \(4x+7=0\). If we solve this, \(x = -7/4 = -1.75\). However, because it is a squared factor, it will have two identical roots.

Key concepts

Quadratic Equation

where \/(a, b\/), and \/(c\/ are coefficients and \/(x\/ is an unknown variable. The solutions to a quadratic equation are the values of \/(x\/ that make the equation true. These solutions are also known as the roots or zeros of the equation.

When tackling these equations, one might encounter one real root, two distinct real roots, or two complex roots, depending on the discriminant \(b^2 - 4ac\). A positive discriminant signifies two distinct real roots, zero means one real root, and a negative discriminant indicates complex roots. Factoring is one of the methods used to find the real roots when possible, and it's particularly straightforward when the quadratic can be expressed as a perfect square.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components or 'factors' that, when multiplied together, give back the original polynomial. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, but not division.

For quadratic equations, the goal of factoring is to decompose the quadratic into a product of two binomials. In the context of our exercise, the equation \(16x^2 + 56x + 49 = 0\) factors neatly into \(\left(4x+7\right)^2\), showcasing a special factoring pattern known as a perfect square trinomial.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(a \eq 0\). If \(a = 0\), the equation would no longer be quadratic, but linear.

Recognizing the standard form is essential as it allows for the application of various methods for solving quadratic equations, such as factoring, using the quadratic formula, completing the square, or graphing. The standard form hints at the properties of the parabola represented by the quadratic, such as its direction of opening, vertex, and axis of symmetry.

Zero Product Property

The zero product property is crucial to understanding why factoring works when solving quadratic equations. It states that if the product of two numbers is zero, then at least one of the multiplicands must be zero. In algebraic terms, if \(a \cdot b = 0\), then \(a = 0\) or \(b = 0\), or both.

This principle underpins step 2 of our exercise, where after factoring the quadratic equation, we set the factor \(4x+7\) equal to zero. Since the entire expression is squared, it reaffirms that \(4x+7\) is indeed zero, leading directly to the solution \(x = -7/4\) or \(x = -1.75\). By interpreting the factored form through the lens of the zero product property, we're able to unlock the values of \(x\) that satisfy the original equation.