Problem 40 - Introductory Algebra for College Students

Question

Solve each quadratic equation by first factoring the perfect square trinomial on the left side. Then apply the square root property. Simplify radicals, if possible. $$y^{2}-14 y+49=18$$

Step-by-Step Solution

Answer

The solution to the equation \(y^{2}-14y+49=18\) is \(y=11,3\).

Step 1: Factor the perfect square trinomial

On the left, the terms can be factored into two identical terms, using the formula, where the sum of the squares \(a^{2}+2ab+b^{2}\) can be written as \((a+b)^{2}\) (the square of the sum of a and b). Therefore, \(y^{2}-14y+49=(y-7)^{2}\). The equation becomes \((y-7)^{2}=18\).

Step 2: Solve using the square root property

The square root property can be used to solve the equation. It states that if \(x^2 = a \), then x is equal to \(\pm \sqrt{a}\). Using this property, set y-7 = to the \(\pm \sqrt{18}\). So it's \(y-7=+\sqrt{18}\) and \(y-7=-\sqrt{18}\).

Step 3: Simplify radicals

We have four as factors of 18 under the square root that can be extracted, giving a simplified radical of \(\sqrt{9}*2=\sqrt{4*4*1}\)=(2*2)=4. The equation therefore translates to \(y-7=+4\) and \(y-7=-4\).

Step 4: Solve for y

Finally, we can solve for y by isolating it, adding 7 to both sides of the equation. We have \(y=+4+7=11\) and \(y=-4+7=3\).

Key Concepts

Factoring

Factoring is a key strategy in solving quadratic equations. When you have a quadratic expression, especially a trinomial, one effective approach to simplify it is to "factor" it. In the given equation \( y^2 - 14y + 49 = 18 \), the terms on the left form a perfect square trinomial. A perfect square trinomial is an expression that can be written as the square of a binomial. In simpler terms, it can be transformed into

• \( (a+b)^2 \) which equals \( a^2 + 2ab + b^2 \)
• or \( (a-b)^2 \) which equals \( a^2 - 2ab + b^2 \)

For example, with \( y^2 - 14y + 49 \), if you look for two identical factors, you'll see it factors into \( (y-7)^2 \). This step simplifies our original equation to \( (y-7)^2 = 18 \). Spotting this pattern is crucial for making the equation easier to manage.

Square Root Property

The square root property is a handy tool that allows us to solve equations involving squared terms directly. Once a quadratic equation is factored into a perfect square form, like \( (y-7)^2 = 18 \), you can apply the square root property. This property tells us that if \( x^2 = a \), then \( x \) is equal to \( \pm \sqrt{a} \).

• Knowing this, we set \( y-7 \) equal to both \( +\sqrt{18} \) and \( -\sqrt{18} \).
• This step splits our equation into two potential solutions: \( y-7 = +\sqrt{18} \) and \( y-7 = -\sqrt{18} \).

Using the square root property is especially helpful because it allows us to handle both positive and negative roots that could fit a squared term, giving a complete set of solutions for equations like the one at hand.

Simplifying Radicals

Simplifying radicals is often the final step in solving such an equation and involves rewriting the radical expression in its simplest form. When working with \( \sqrt{18} \), we aim to express it with simpler components. To simplify \( \sqrt{18} \), you need to express 18 as the product of perfect squares. Notably, 18 is \( 9 \times 2 \).

• This allows \( \sqrt{18} \) to be rewritten as \( \sqrt{9 \times 2} \),
• which then simplifies to \( \sqrt{9} \times \sqrt{2} \).
• The square root of 9 is 3, giving you \( 3\sqrt{2} \).

Therefore, \( \pm \sqrt{18} \) becomes \( \pm 3\sqrt{2} \).
Simplifying radicals helps make your final solutions more manageable and presentable. Returning to our equation, once the radicals are simplified, the steps to solve for \( y \) become more straightforward, as we can easily add or subtract these simplified terms.