Problem 17 - Precalculus Student Solutions Manual 5th

Question

Solve each equation by the zero-factor property. $$-4 x^{2}+x=-3$$

Step-by-Step Solution

Answer

x = \frac{3}{4} or x = -1

Step 1 - Move all terms to one side

Transform the equation so that all terms are on one side, setting it equal to zero. Given the equation $$-4x^2 + x = -3$$, add 3 to both sides to get: $$-4x^2 + x + 3 = 0$$

Step 2 - Factor the quadratic equation

Factor the quadratic equation $$-4x^2 + x + 3 = 0$$. We need to find two binomials such that their product equals the given quadratic expression. Factoring gives us: $$(4x - 3)(x + 1) = 0$$

Step 3 - Apply the Zero-Factor Property

According to the zero-factor property, if the product of two factors equals zero, then at least one of the factors must be zero. Set each factor equal to zero and solve for \(x\): $$4x - 3 = 0$$ $$x + 1 = 0$$

Step 4 - Solve each equation

Solve each equation separately to find the values of \(x\): For $$4x - 3 = 0$$, add 3 to both sides and then divide by 4: $$4x = 3$$ $$x = \frac{3}{4}$$ For $$x + 1 = 0$$, subtract 1 from both sides: $$x = -1$$

Key Concepts

quadratic equation

A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually denoted as x) is 2. Quadratic equations generally take the form:

\[ax^2 + bx + c = 0\]
where \(a\), \(b\), and \(c\) are constants, with \(a\) not equal to zero. In the example provided, the equation
\(-4x^2 + x + 3 = 0\)
is a quadratic equation because it has the variable \(x\) raised to the second power.

factoring

Factoring in algebra involves breaking down an expression into simpler expressions that can be multiplied together to reproduce the original expression. For quadratic equations, this often means finding two binomial expressions whose product equals the quadratic expression.
In our example, we need to factor
\(-4x^2 + x + 3 = 0\).
This can be broken down into
\((4x - 3)(x + 1) = 0\).
Factoring is a crucial step because it allows us to use the zero-factor property.

solving equations

Solving equations involves finding the values of the variable that make the equation true. In our example, we use the zero-factor property to solve for \(x\). According to this property, if the product of two factors is zero, then at least one of the factors must be zero.
Therefore, for
\((4x - 3)(x + 1) = 0\),
either
\(4x - 3 = 0\)
or
\(x + 1 = 0\)
must be true.
Solving these equations separately:
For
\(4x - 3 = 0\), add 3 to both sides to get
\(4x = 3\).
Then, divide by 4 to get
\(x = \frac{3}{4}\).
For
\(x + 1 = 0\), subtract 1 from both sides to get
\(x = -1\).
So, the solutions to the equation are
\(x = \frac{3}{4}\)
and
\(x = -1\).