Question

How do you write in standard form $y=\left( x-1 \right)\left( 5x+2 \right)$?

Answer 1

Hint: Now the given equation is in factored form. To write it in general form we will use the distributive property in the equation. Hence we will simplify the equation and write it in the form of equation $a{{x}^{2}}+bx+c=0$ . Hence we can easily write the given equation in general form.

Complete step by step solution:
Now the given expression is factored form a quadratic equation.
Hence we can see that (x – 1) and (5x + 2) are the factors of the equation.
Now to find the actual equation we will multiply the factors using the distributive property.
Now we know that according to distributive property we have $c\left( a+b \right)=ca+cb$
Hence using this property we get,
$\Rightarrow y=\left( x-1 \right)5x+\left( x-1 \right)2$
Now we can use commutative property of multiplication which says $a.b=b.a$ Hence we get,
$\Rightarrow y=5x\left( x-1 \right)+2\left( x-1 \right)$
Now again using distributive property we get,
$\Rightarrow y=5{{x}^{2}}-5+2x-2$
Now simplifying the equation we get the value of y as,
$\Rightarrow y=5{{x}^{2}}+2x-7$
Now we can see that the equation obtained is a quadratic equation in the form $a{{x}^{2}}+bx+c$ where a = 5, b = 2 and c = -7.
Hence the given equation is written in general form as $5{{x}^{2}}+2x-7$

Note: Now note that from the given factors of the equation we can easily find the roots of the equation. For each factor of the form $ax+b=0$ we will solve the equation and find the root of the equation. Hence we can get both the roots of the quadratic equation. Now from the obtained roots we can find the sum of the roots and product of the roots. Now we know that the quadratic equation is given as ${{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta =0$ where $\alpha ,\beta $ are the roots of the equation.