(ii) Factorise [tex]$6 y^2-109 y-95$[/tex]. (iii) Find the value of [tex]$y$[/tex].
Answer (1)
Factor the quadratic expression 6 y 2 − 109 y − 95 by finding two numbers that multiply to 6 × − 95 = − 570 and add up to − 109 , which are − 114 and 5 .
Rewrite the expression as 6 y 2 − 114 y + 5 y − 95 and factor by grouping to get ( 6 y + 5 ) ( y − 19 ) .
Set each factor to zero: 6 y + 5 = 0 or y − 19 = 0 .
Solve for y to find the two solutions: y = − 6 5 and y = 19 . Thus, the values of y are − 6 5 , 19 .
Explanation
Understanding the Problem We are given the quadratic expression 6 y 2 − 109 y − 95 and asked to factorise it and find the values of y for which the expression equals zero. This means we need to solve the quadratic equation 6 y 2 − 109 y − 95 = 0 .
Finding the Right Numbers To factorise the quadratic expression, we look for two numbers whose product is 6 × − 95 = − 570 and whose sum is − 109 . These numbers are − 114 and 5 , since − 114 × 5 = − 570 and − 114 + 5 = − 109 .
Rewriting the Middle Term Now we rewrite the middle term of the quadratic expression using these two numbers:
6 y 2 − 109 y − 95 = 6 y 2 − 114 y + 5 y − 95 .
Factoring by Grouping Next, we factor by grouping:
6 y 2 − 114 y + 5 y − 95 = 6 y ( y − 19 ) + 5 ( y − 19 ) .
Final Factorisation We factor out the common term ( y − 19 ) :
6 y ( y − 19 ) + 5 ( y − 19 ) = ( 6 y + 5 ) ( y − 19 ) .
So, the factorised form of the quadratic expression is ( 6 y + 5 ) ( y − 19 ) .
Setting Factors to Zero To find the values of y , we set the factorised expression equal to zero:
( 6 y + 5 ) ( y − 19 ) = 0 .
This means either 6 y + 5 = 0 or y − 19 = 0 .
Solving for y (1) Solving 6 y + 5 = 0 for y , we get:
6 y = − 5
y = − 6 5 .
Solving for y (2) Solving y − 19 = 0 for y , we get:
y = 19 .
Therefore, the values of y are − 6 5 and 19 .
Final Answer The factorised form of the quadratic expression is ( 6 y + 5 ) ( y − 19 ) , and the values of y are − 6 5 and 19 .
Examples
Quadratic equations are not just abstract math; they appear in various real-world scenarios. For instance, when designing a bridge, engineers use quadratic equations to calculate the curve of an arch. Similarly, in physics, the trajectory of a projectile, like a ball thrown in the air, can be modeled using a quadratic equation. Understanding how to factorize and solve these equations helps in predicting outcomes and designing structures effectively.
