Question

Solve the quadratic equation by finding square roots or by using the quadratic formula. Explain why you chose the method. $$4 y^{2}-49=0$$

Short answer

The solutions to the equation are \(y = 3.5\) and \(y = -3.5\).

Step-by-step solution

Rewrite the Equation to Isolate \(y^2\)

Begin by adding 49 to both sides of the equation to isolate \(y^2\). The equation becomes \(4y^2 = 49\).

Solve for \(y^2\)

Continue by dividing both sides by 4, so that \(y^2\) is isolated on one side of the equation. This gives \(y^2 = 12.25\).

Solve for \(y\)

Finally, take the square root of both sides of the equation. This gives two solutions, as \(\sqrt{y^2} = \sqrt{12.25}\) means that \(y\) is either +3.5 or -3.5. Therefore, \(y = 3.5\) or \(y = -3.5\).

Key concepts

Finding Square Roots

When we solve a quadratic equation such as \(4y^2 - 49 = 0\), we can sometimes use the approach of finding square roots. This method is particularly handy when the equation is already in a form that isolates a squared term on one side.

To go about this, we perform operations that aim to leave the squared variable by itself, with no other numbers or variables attached to it. In the given exercise, we start by adding 49 to both sides to get \(4y^2 = 49\). Next, we divide everything by 4, simplifying it to \(y^2 = 12.25\).

At this point, we look for the square root of each side, knowing that square roots undo the squaring operation; thus, \(\sqrt{y^2} = \sqrt{12.25}\) gives us two possible values for \(y\): +3.5 or -3.5 since both 3.5 squared and -3.5 squared equal 12.25. We take both results because squaring a positive or negative number yields a positive result, so without prior information, both are viable solutions.

Quadratic Formula

The quadratic formula is a powerful and universal tool for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). It is derived from the process of completing the square and gives the exact solutions for the values of \(x\) that satisfy the equation. The formula itself is given by \[-b \pm \sqrt{b^2 - 4ac}\] over \(2a\).

The discriminant, \(b^2 - 4ac\), inside the square root is the key to determining the nature of the roots: if it's positive, there are two real solutions; if it's zero, there is one real solution; and if it's negative, it indicates complex solutions.

For the given exercise, we could apply the quadratic formula by setting \(a = 4\), \(b = 0\), and \(c = -49\). However, in this case, finding square roots is more straightforward since the equation is already in an easily reducible form. This demonstrates how critical it is to evaluate the best method to apply based on the specific quadratic equation at hand.

Isolating Variables

Isolating variables is a fundamental skill in algebra that involves manipulating an equation to leave the variable of interest alone on one side of the equation. We use this technique extensively when solving for unknowns.

In the step-by-step solution, isolating the variable begins by rearranging the equation so that \(y^2\) is by itself. Here's what it involves: moving anything that is not part of the \(y^2\) term to the other side using addition or subtraction and then using multiplication or division to undo any coefficients attached to the squared term.

Importance of Isolating \(y^2\)

By isolating the term \(y^2\), we simplify the equation down to a basic form, which then allows us to use simpler operations to find the solution—specifically, extracting the square root. Without isolating \(y^2\) first, we cannot directly apply the square root operation, as it would impact other terms in the equation.

Ultimately, the goal of isolating the variable is to peel away all of the layers of operations affecting it so that we can discover the value of the variable itself.