Problem 47 - Algebra 1

Question

Write the quadratic equation in standard form. Solve using the quadratic formula. $$5 x-1=-6 x^{2}$$

Step-by-Step Solution

Answer

The roots of the quadratic equation are \( x1 = -1/6 \) and \( x2 = -1 \).

Step 1: Rearrange the equation into standard form

Starting with the equation: \( 5x - 1 = -6x^2 \), the first step is to bring all the terms on one side, which results in the standard form. The standard form is \( 6x^2 + 5x - 1 = 0 .\) Here, a = 6, b = 5, and c = -1.

Step 2: Calculate the discriminant

Use the values of a, b and c to calculate the discriminant. The discriminant \( D = b^2 - 4ac = (5)^2 - 4*6*(-1) = 25 + 24 = 49\).

Step 3: Solve using the Quadratic Formula

Substitute the values a = 6, b = 5 and D = 49 into the quadratic formula. You get two possible solutions for x: \( x1 = (-5 + \sqrt{49}) / (2*6) = -1/6\), and \( x2 = (-5 - \sqrt{49}) / (2*6) = -1 \).

Key Concepts

Standard Form of a Quadratic Equation

Understanding the standard form of a quadratic equation is crucial when you're tackling algebra problems. It is the baseline from which all further calculations and understanding of the equation's characteristics are derived.

The standard form of a quadratic equation looks like this: \( ax^2 + bx + c = 0 \) Here, \( x \) represents the variable, while \( a \) , \( b \) , and \( c \) represent known numbers where \( a \) is not equal to zero. If \( a \) were zero, the equation would no longer be quadratic, but linear.

• \( a \) is the coefficient of the squared term (\( x^2 \) term).
• \( b \) is the coefficient of the \( x \) term.
• \( c \) is the constant term.

Setting an equation to this form simplifies the process of analyzing and solving quadratic equations, using various methods including factoring, completing the square, or applying the quadratic formula.

Quadratic Formula

The quadratic formula is a widely-used tool for solving quadratic equations that don't readily factor or could be complicated to complete the square. This powerful formula is derived from the process of completing the square and is given by:

\( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \) where \( a \) , \( b \) , and \( c \) are the same coefficients from the standard form of the quadratic equation. The symbol \( \pm \) indicates that there will generally be two solutions for \( x \) since a square root has both a positive and negative value.

Applying the Quadratic Formula

To solve a quadratic equation using the quadratic formula, you just need to plug the values of \( a \) , \( b \) , and \( c \) into the formula and simplify. This process will give you the equation's roots, which are the solutions for \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \) .

Discriminant Calculation

To predict the nature of solutions for a quadratic equation without actually solving it, you can calculate the discriminant. The discriminant, often denoted as \( D \) , is the part of the quadratic formula under the square root, \( b^2 - 4ac \) .

It tells us:

• If \( D > 0 \) , there are two real and distinct solutions.
• If \( D = 0 \) , there is exactly one real solution (repeated).
• If \( D < 0 \) , there are no real solutions, but two complex solutions.

Knowing the value of \( D \) can also help you determine the number of times the graph of the quadratic equation will intersect the x-axis. In the original exercise provided, \( D\) was calculated as 49, indicating that the equation has two distinct real solutions. This is because a positive \( D\) means that \( \sqrt{D} \) is a real number, creating two different possible solutions when applied to the \( \pm \) in the quadratic formula.