Problem 25 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by factoring. $$4 x=3-7 x^{2}$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{3}{7}\) and \(x = -1\).

Step 1: Bring All Terms to One Side

The given equation is \(4x = 3 - 7x^2\). To start solving it, we need to change it to the form of \(ax^2 + bx + c = 0\). Move all terms to one side of the equation to get: \(-7x^2 - 4x + 3 = 0\).

Step 2: Standard Form

The equation \(-7x^2 - 4x + 3 = 0\) is now in standard quadratic form. To help with factoring, reorder it as \(-7x^2 - 4x + 3 = 0\).

Step 3: Factor the Quadratic

To factor the quadratic equation \(-7x^2 - 4x + 3 = 0\), we need two numbers that multiply to \(a imes c = -21\) and add up to \(-4\). These numbers are \(-7\) and \(3\). The equation factors to \((-7x + 3)(x + 1) = 0\).

Step 4: Solve for x

Using the zero product property, set each factor equal to zero: \(-7x + 3 = 0\) and \(x + 1 = 0\). Solve \(-7x + 3 = 0\): \(-7x = -3\) then \(x = \frac{3}{7}\). Solve \(x + 1 = 0\): \(x = -1\). Therefore, the solutions are \(x = \frac{3}{7}\) and \(x = -1\).

Key Concepts

Factoring

Factoring is a powerful method used to simplify polynomial equations, especially quadratic equations, to find their roots. It involves expressing the polynomial as a product of its factors. For quadratic equations, the goal is to rewrite the equation in a form of two binomials that, when multiplied together, give the original quadratic expression.
A typical quadratic equation has the form \(ax^2 + bx + c = 0\). To factor this equation, we look for two numbers that multiply to \(a \times c\) and add up to \(b\). Once we find these numbers, we can decompose the middle term \(bx\) into two terms using these numbers and then factor by grouping.
Factoring is especially convenient because it allows us to use other methods like the zero product property to solve the equation efficiently.

Zero Product Property

The Zero Product Property is a fundamental principle that helps solve equations expressed as a product of factors set to zero. According to this property, if the product of two numbers is zero, then at least one of the numbers must be zero. This means that for an equation in the form \((m)(n) = 0\), either \(m = 0\), \(n = 0\), or both.
Applying the zero product property to a factored quadratic equation allows us to find the value of \(x\) by setting each factor equal to zero and solving the resulting simple equations.
In the example problem, the equation \((-7x + 3)(x + 1) = 0\) implies \(-7x + 3 = 0\) or \(x + 1 = 0\). Solving these two equations separately gives the solutions for \(x\). This property makes it straightforward to obtain the roots of the polynomial.

Standard Quadratic Form

Standard Quadratic Form is the commonly recognized format of a quadratic equation, written as \(ax^2 + bx + c = 0\). This form is essential because it allows easier manipulation and application of various solving techniques like factoring, completing the square, or using the quadratic formula.
One of the first steps in solving a quadratic equation is to convert any given equation into the standard form. This often involves moving all terms to one side of the equation so that the expression equals zero.
For our original equation, \(4x = 3 - 7x^2\), rearrangement was necessary to achieve the form \(-7x^2 - 4x + 3 = 0\). Once in standard form, the equation becomes more straightforward to manage, enabling the use of factoring techniques to find the equation's roots efficiently.