Question

solve by applying the zero product\
ewline $12t(t + 2) = t - 10$\
ewline if there is more than one solution, s

Explanation:

Step1: Expand and rearrange the equation

First, expand the left - hand side of the equation \(12t(t + 2)=t-10\). Using the distributive property \(a(b + c)=ab+ac\), we have \(12t\times t+12t\times2=t - 10\), which simplifies to \(12t^{2}+24t=t - 10\).
Then, move all terms to the left - hand side to get a quadratic equation in standard form \(ax^{2}+bx + c = 0\). Subtract \(t\) from both sides and add 10 to both sides: \(12t^{2}+24t - t+10 = 0\), so \(12t^{2}+23t + 10=0\).

Step2: Factor the quadratic equation (or use quadratic formula)

We can try to factor the quadratic \(12t^{2}+23t + 10\). We need two numbers that multiply to \(12\times10 = 120\) and add up to 23. The numbers are 8 and 15.
Rewrite the middle term: \(12t^{2}+8t+15t + 10 = 0\).
Group the terms: \((12t^{2}+8t)+(15t + 10)=0\).
Factor out the greatest common factor from each group: \(4t(3t + 2)+5(3t + 2)=0\).
Now, factor out the common binomial factor \((3t + 2)\): \((4t + 5)(3t+2)=0\).

Step3: Apply the zero - product property

The zero - product property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\).
So, we set each factor equal to zero:

• For \(4t+5 = 0\), subtract 5 from both sides: \(4t=-5\), then divide by 4: \(t=-\frac{5}{4}\).
• For \(3t + 2=0\), subtract 2 from both sides: \(3t=-2\), then divide by 3: \(t=-\frac{2}{3}\).

Answer:

The solutions are \(t =-\frac{5}{4}\) and \(t=-\frac{2}{3}\)