Problem 63 - Algebra 1

Question

The solution of a quadratic equation can be found by graphing each side separately and locating the points of intersection. You may wish to consult page 532 for help in approximating solutions. $$ 0.75 x^{2}+2.67 x=6.22 x^{2}-4.1 $$

Step-by-Step Solution

Answer

The solutions to the equation are \( x = 0.28 \) and \( x = -2.97 \).

Step 1: Simplify the equation to standard form

Simplify the given equation to the standard form \( ax^{2} + bx + c = 0 \). Start by bringing all the terms to one side, so the equations becomes \( 0.75x^{2} + 2.67x - 6.22x^{2} + 4.1 = 0 \). Then combine like terms, which gives \( -5.47x^{2} + 2.67x + 4.1 = 0 \).

Step 2: Solve the simplified equation

Use the quadratic formula to solve the equation. The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Plugging the coefficients into the quadratic formula gives \( x = \frac{-2.67 \pm \sqrt{(2.67)^2 - 4*(-5.47)*4.1}}{2*-5.47} \)

Step 3: Calculate the solutions

Calculate the values under the square root and simplify the expressions to find the two roots of the equation. The solutions will be \( x = 0.28 \) and \( x = -2.97 \). However, it's crucial to check these solutions by substituting them back into the original equation to see whether they hold true.

Key Concepts

Solving Quadratic Equations

Solving quadratic equations can be approached through various methods, with each offering its unique pathway to find the values of \(x\) that satisfy the equation.
The most common techniques include factoring, completing the square, and using the quadratic formula. However, before applying any of these methods, it's often helpful to rearrange the given equation into a standard format. This usually means moving all terms to one side to allow the equation to equal zero.
By solving for \(x\), you're essentially finding the roots of the equation, which are the points where the graph of the equation intersects the \(x\)-axis. Understanding these concepts is fundamental for building more advanced math skills.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, particularly when other methods like factoring aren't feasible. This formula is written as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) and is derived from the process of completing the square.
The formula requires the coefficients \(a\), \(b\), and \(c\) from the standard form of the equation, \( ax^2 + bx + c = 0 \). By substituting these coefficients into the formula, you can determine the roots of the quadratic equation.
The expression under the square root, \(b^2 - 4ac\), is known as the discriminant. It reveals valuable information about the roots, such as whether they are real or complex numbers. Utilizing the quadratic formula thus provides a meticulous way of obtaining a solution.

Graphing Equations

Understanding how to graph equations is a crucial aspect of solving quadratic equations. By visualizing the equation on a graph, you can see where it intersects the \(x\)-axis, which corresponds to the roots found algebraically.
For quadratic equations, the graph forms a parabola, which can open upwards or downwards depending on the sign of the leading coefficient \(a\). To graph a quadratic equation, it's helpful to pinpoint the vertex, axis of symmetry, and intercepts.
Additionally, graphing provides an excellent opportunity to verify solutions obtained through algebraic methods, offering a visual confirmation of the intersection points.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is expressed as \( ax^2 + bx + c = 0 \). Getting an equation into this form is often the first step in solving it, as it sets the stage for applying methods like the quadratic formula.
Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The coefficient \(a\) determines the direction and width of the parabola on the graph, while \(b\) and \(c\) impact the location and intersection points.
Simplifying an equation to its standard form can also reveal opportunities for factoring, especially when \(b\) or \(c\) is an accommodating value. This structural simplification is a vital skill for handling more complex quadratic equations as you advance in mathematics.