Solving Quadratics Tutorial 11g Relating to the Real World Members of the science club launch a model rocket from ground level with a starting velocity of 96 feet per second. After how many seconds will the rocket have an altitude of 128 ft? The quadratic formula is important in physics when finding vertical motion. When an object is dropped, thrown, or launched either straight up or down, you can use the vertical motion formula to find the height of the object. Vertical motion formula: h = -16t2 + vt +s Introduction The vertical motion formula is a quadratic equation. A quadratic equation is any equation that can be written in the form ax2 + bx + c = 0, where a, b, and c are real numbers and a 0. In this lesson you will learn how to solve any quadratic by using the quadratic formula. Later in this lesson we will take a look back at the previous vertical motion problem and learn how to solve it. The Quadratic Formula: If ax2 + bx +c = 0 and a 0, then: b b 4ac x 2a 2 Remember: The solution of ax2 + bx +c = 0 are numbers we call roots of the equation. Also, the solution set of y = ax2 + bx +c is a set of ordered pairs of numbers that satisfy the function. The Quadratic Formula: cont. . . If ax2 + bx +c = 0 and a 0, then: b b 4ac x 2a 2 As you can see above, the coefficients of a quadratic equation (values of a, b, & c) are used to solve the equation. Example 1 Use the Quadratic Formula to solve: x2 + 5x = -6= -6 x2 + 5x = -6. First we must put the equation in standard form. Identify the values of a, b, & c: Remember ax2 + bx + c = 0 Substitute 1 for a, 5 for b, and 6 for c in the Quadratic Formula. Then simplify the resulting expression. CLICK HERE! +6 =+6 5x + 6 6=0 1x2 + 5 a = 1, 1 b=5 5, c = 66 b b 4ac x 2a 2 5 5 4 1 6 x 2 1 2 Example 1 Simplify: Use the Quadratic Formula to solve: x2 + 5x = -6. Cont . . . 5 5 4 1 6 x 2 1 2 5 25 24 x 2 1 5 1 x 2 1 5 1 x 2 or 5 1 x 2 x 2 or x 3 Relating to the Real World Members of the science club launch a model rocket from ground level with a starting velocity of 96 feet per second. After how many seconds will the rocket have an altitude of 128 ft? We will need to use the vertical motion formula to find the height of the object. Vertical motion formula: h = -16t2 + vt + s h is the height of the object in feet. t is the time it takes an object to rise or fall to a given height. v is the starting velocity in feet per second. s is the starting height. Relating to the Real World Members of the science club launch a model rocket from ground level with a starting velocity of 96 feet per second. After how many seconds will the rocket have an altitude of 128 ft? Vertical motion formula: h = -16t2 + vt + s h is the height of the object in feet. t is the time it takes an object to rise or fall to a given height. v is the starting velocity in feet per second. s is the starting height. h = -16t2 + vt + s 128 = -16t2 + 96t + 0 Use the Vertical motion formula. Substitute 128 for h, 96 for v, and 0 for s. 0 = -16t2 + 96t –128 Subtract 128 from each side. Members of the science club launch a model rocket from ground level with a starting velocity of 96 feet per second. After how many seconds will the rocket have an altitude of 128 ft? Relating to the Real World 0 = -16t2 + 96t –128 h = -16t2 + vt + s 128 = -16t2 + 96t + 0 Use the Vertical motion formula. Substitute 128 for h, 96 for v, and 0 for s. 0 = -16t2 + 96t –128 Subtract 128 from each side. Members of the science club launch a model rocket from ground level with a starting velocity of 96 feet per second. After how many seconds will the rocket have an altitude of 128 ft? Relating to the Real World 0 = -16t2 + 96t –128 b b 2 4ac t 2a 96 962 4(16)(128) t 2(16) 96 9216 8192 t 32 96 1024 t 32 96 1024 t 32 96 1024 or t 32 To solve, use the quadratic formula. Substitute –16 for a, 96 for b, and –128 for c. Simplify Write two solutions and then simplify. Members of the science club launch a model rocket from ground level with a starting velocity of 96 feet per second. After how many seconds will the rocket have an altitude of 128 ft? Relating to the Real World 0 = -16t2 + 96t –128 b b 2 4ac t 2a 96 962 4(16)(128) t 2(16) 96 9216 8192 t 32 96 1024 t 32 t 96 1024 32 or t 96 1024 32 96 1024 32 96 32 t 32 t 96 1024 32 96 32 or t 32 or t t 2 or t 4 The rocket is 128 ft. off the ground after 2 seconds & after 4 seconds.

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