Solving Reducible Equations
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Question 1 of 5
1. Question
Solve for `x``3^(2x)-3^x-72=0`Hint
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Reducible equations are non-quadratic equations that can be reduced into a quadratic equation for easier solving.First, rewrite the equation as a quadratic equation by assigning a new variablelet`u=3^x``u^2=3^(2x)``3^(2x)-3^x-72` `=` `0` `u^2-u-72` `=` `0` Substitute new variable Solve for `u` by using cross method.
`(u-9)(u+8)` `=` `0` `u-9` `=` `0` `u-9` `+9` `=` `0` `+9` `u` `=` `9` `u+8` `=` `0` `u+8` `-8` `=` `0` `-8` `u` `=` `-8` Finally, substitute `u=3^x` to get the values of `x``u` `=` `9` `3^x` `=` `9` Substitute `u=3^x` `3^x` `=` `3^2` `x` `=` `2` Equal bases means equal exponents `u` `=` `-8` `3^x` `=` `-8` Substitute `u=3^x` This has no solution since there is no `x` value that can make `3^x` negative`x=2` -
Question 2 of 5
2. Question
Solve for `x``2^(2x)-3.2^x-40=0`Hint
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Reducible equations are non-quadratic equations that can be reduced into a quadratic equation for easier solving.First, rewrite the equation as a quadratic equation by assigning a new variablelet`u=2^x``2^(2x)-3.2^x-40` `=` `0` `u^2-3u-40` `=` `0` Substitute new variable Solve for `u` by using cross method.
`(u+5)(u-8)` `=` `0` `u+5` `=` `0` `u+5` `-5` `=` `0` `-5` `u` `=` `-5` `u-8` `=` `0` `u-8` `+8` `=` `0` `+8` `u` `=` `8` Finally, substitute `u=2^x` to get the values of `x``u` `=` `8` `2^x` `=` `8` Substitute `u=2^x` `2^x` `=` `2^3` `x` `=` `3` Equal bases means equal exponents `u` `=` `-5` `2^x` `=` `-5` Substitute `u=2^x` This has no solution since there is no `x` value that can make `2^x` negative`x=3` -
Question 3 of 5
3. Question
Solve for `x``x^4-7x^2-18=0`Hint
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Reducible equations are non-quadratic equations that can be reduced into a quadratic equation for easier solving.First, rewrite the equation as a quadratic equation by assigning a new variablelet`u=x^2``x^4-7x^2-18` `=` `0` `u^2-7u-18` `=` `0` Substitute new variable Solve for `u` by using cross method.
`(u-9)(u+2)` `=` `0` `u-9` `=` `0` `u-9` `+9` `=` `0` `+9` `u` `=` `9` `u+2` `=` `0` `u+2` `-2` `=` `0` `-2` `u` `=` `-2` Finally, substitute `u=x^2` to get the values of `x``u` `=` `9` `x^2` `=` `9` Substitute `u=x^2` `sqrt(x^2)` `=` `sqrt9` Get the square root of both sides `x` `=` `+-3` `u` `=` `-2` `x^2` `=` `-2` Substitute `u=x^2` This has no solution since there is no `x` value that can make `x^2` negative`x=3, -3` -
Question 4 of 5
4. Question
Solve for `x``4x^4+3x^2-10=0`Hint
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Excellent!
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Reducible equations are non-quadratic equations that can be reduced into a quadratic equation for easier solving.First, rewrite the equation as a quadratic equation by assigning a new variablelet`u=x^2``4x^4+3x^2-10` `=` `0` `4u^2+3u-10` `=` `0` Substitute new variable Solve for `u` by using cross method.
`(4u-5)(u+2)` `=` `0` `4u-5` `=` `0` `4u-5` `+5` `=` `0` `+5` `4u` `=` `5` `u` `=` `5/4` `u+2` `=` `0` `u+2` `-2` `=` `0` `-2` `u` `=` `-2` Finally, substitute `u=x^2` to get the values of `x``u` `=` `5/4` `x^2` `=` `5/4` Substitute `u=x^2` `sqrt(x^2)` `=` `sqrt(5/4)` Get the square root of both sides `x` `=` `+-(sqrt5)/2` `u` `=` `-2` `x^2` `=` `-2` Substitute `u=x^2` This has no solution since there is no `x` value that can make `x^2` negative`x=(sqrt5)/2, -(sqrt5)/2` -
Question 5 of 5
5. Question
Solve for `x``4^x+3*2^x-10=0`Hint
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Fantastic!
Incorrect
Reducible equations are non-quadratic equations that can be reduced into a quadratic equation for easier solving.First, rewrite the equation as a quadratic equation by assigning a new variablelet`u=2^x``u^2=2^(2x)``4^x+3*2^x-10` `=` `0` `(2^2)^x+3*2^x-10` `=` `0` `2^(2x)+3*2^x-10` `=` `0` `u^2+3u-10` `=` `0` Substitute new variable Solve for `u` by using cross method.
`(u-2)(u+5)` `=` `0` `u-2` `=` `0` `u-2` `+2` `=` `0` `+2` `u` `=` `2` `u+5` `=` `0` `u+5` `-5` `=` `0` `-5` `u` `=` `-5` Finally, substitute `u=2^x` to get the values of `x``u` `=` `2` `2^x` `=` `2` Substitute `u=2^x` `2^x` `=` `2^1` `x` `=` `1` Equal bases means equal exponents `u` `=` `-5` `2^x` `=` `-5` Substitute `u=2^x` This has no solution since there is no `x` value that can make `3^x` negative`x=1`
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- Sum & Product of Roots 1
- Sum & Product of Roots 2
- Sum & Product of Roots 3
- Sum & Product of Roots 4
- Solving Equations by Factoring 1
- Solving Equations Using the Quadratic Formula
- Completing the Square 1
- Completing the Square 2
- Intro to Quadratic Functions (Parabolas) 1
- Intro to Quadratic Functions (Parabolas) 2
- Intro to Quadratic Functions (Parabolas) 3
- Graph Quadratic Functions in Standard Form 1
- Graph Quadratic Functions in Standard Form 2
- Graph Quadratic Functions by Completing the Square
- Graph Quadratic Functions in Vertex Form
- Write a Quadratic Equation from the Graph
- Write a Quadratic Equation Given the Vertex and Another Point
- Quadratic Inequalities 1
- Quadratic Inequalities 2
- Quadratics Word Problems 1
- Quadratics Word Problems 2
- Quadratic Identities
- Graphing Quadratics Using the Discriminant
- Positive and Negative Definite
- Applications of the Discriminant 1
- Applications of the Discriminant 2
- Solving Reducible Equations