Factor quadratic expressions:
2x^2 + 9x + 4
10x^2 + 70x + 100
Solve quadratic equations by factoring:
2x^2 + 9x + 4 = 0
Solve by taking square roots:
(x - 5)^2 - 81 = 0
2x^2 + 9x + 4 = 0
(2x+1)(x+4)=0
x=-1/2, -4
(x - 5)^2 - 81 = 0
x-5=+- 9
x=14, -4
2nd:
10x^2 + 70x + 100
= 10(x^2 + 7x + 10)
= 10(x+5)(x+2)
3rd:
2x^2 + 9x + 4 = 0
(2x + 1)(x + 4) = 0
carry on
4th:
real easy ...
(x-5)^2 - 81 = 0
(x-5)^2 = 81
take √ of both sides
x-5= ±9
x = 5 ± 9
= 14 or -4
To factor quadratic expressions:
1. 2x^2 + 9x + 4:
To factor this quadratic expression, we need to find two numbers that multiply to give 8 (2x4) and add up to give 9. The numbers are 1 and 8.
So, the factored form of the expression is (2x + 1)(x + 4).
2. 10x^2 + 70x + 100:
To factor this quadratic expression, we can factor out the common factor of 10.
So, the factored form of the expression is 10(x^2 + 7x + 10).
Now, we can factor the quadratic expression inside the parentheses by finding two numbers that multiply to give 10 and add up to give 7. The numbers are 2 and 5.
So, the final factored form is 10(x + 2)(x + 5).
To solve quadratic equations by factoring:
1. 2x^2 + 9x + 4 = 0:
First, we need to factor the quadratic expression. From the previous step, we know that the factored form of the expression is (2x + 1)(x + 4).
Now, we set each factor equal to zero:
2x + 1 = 0 --> 2x = -1 --> x = -1/2
x + 4 = 0 --> x = -4
Therefore, the solutions to the quadratic equation are x = -1/2 and x = -4.
To solve by taking square roots:
1. (x - 5)^2 - 81 = 0:
First, simplify the equation:
(x - 5)^2 - 81 = 0
(x - 5)^2 = 81
Now, take the square root of both sides:
x - 5 = ±√81
x - 5 = ±9
Now, solve for x:
x = 5 + 9 or x = 5 - 9
x = 14 or x = -4
Therefore, the solutions to the quadratic equation are x = 14 and x = -4.
To factor quadratic expressions, you need to find two binomials that, when multiplied together, give you the original expression. Here's how you can approach factoring the given quadratic expressions:
1. 2x^2 + 9x + 4:
To factor this expression, you need to find the two numbers that multiply to give you 2 * 4 = 8 and add up to give you 9 (the coefficient of the middle term). The numbers that satisfy this condition are 1 and 8. Therefore, you can factor the expression as follows:
2x^2 + 9x + 4 = (2x + 1)(x + 4)
2. 10x^2 + 70x + 100:
You can start by factoring out the greatest common factor (GCF) from all three terms, which in this case, is 10. Then, you need to factor the trinomial expression. Divide each term by 10 to get:
10x^2 + 70x + 100 = 10(x^2 + 7x + 10)
To further factor the trinomial, you need to find two numbers that multiply to give you 1 * 10 = 10 and add up to give you 7. The numbers that satisfy this condition are 2 and 5. Therefore, you can factor the trinomial as follows:
10x^2 + 70x + 100 = 10(x + 2)(x + 5)
Moving on to solving quadratic equations by factoring:
3. 2x^2 + 9x + 4 = 0:
To solve this equation by factoring, you need to factor the quadratic expression on the left side and set each factor equal to zero. Using the factors we found in the first example:
(2x + 1)(x + 4) = 0
Now, set each factor equal to zero and solve for x:
2x + 1 = 0 => 2x = -1 => x = -1/2
x + 4 = 0 => x = -4
Therefore, the solutions to the quadratic equation are x = -1/2 and x = -4.
Next, solving equations by taking square roots:
4. (x - 5)^2 - 81 = 0:
To solve this equation by taking square roots, you want to isolate the square term first. Add 81 to both sides of the equation:
(x - 5)^2 = 81
Now, take the square root of both sides, considering both positive and negative square roots:
x - 5 = ±√81
x - 5 = ±9
Now, solve for x by adding 5 to both sides of the equation:
x = 5 ± 9
Therefore, the solutions to the quadratic equation are x = 14 and x = -4.