{"id":2129,"date":"2020-07-28T08:53:19","date_gmt":"2020-07-28T08:53:19","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&p=2129"},"modified":"2020-07-28T10:10:05","modified_gmt":"2020-07-28T10:10:05","slug":"notes-quadratics","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/quadratics\/notes-quadratics\/","title":{"rendered":"Notes – Quadratics"},"content":{"rendered":"

Roots of the equation\u00a0 \r\na<\/mi>x<\/mi>2<\/mn><\/msup>+<\/mo>b<\/mi>x<\/mi>+<\/mo>c<\/mi>=<\/mo>0<\/mn><\/math>\r\n <\/strong><\/span><\/p>\r\n

Multiplying the equation\u00a0 \r\na<\/mi>x<\/mi>2<\/mn><\/msup>+<\/mo>b<\/mi>x<\/mi>+<\/mo>c<\/mi>=<\/mo>0<\/mn><\/math>\r\n \u00a0\u00a0both sides by\u00a0 \r\n4<\/mn>a<\/mi><\/math>\r\n , we get<\/p>\r\n

\r\n4<\/mn>a<\/mi>2<\/mn><\/msup>x<\/mi>2<\/mn><\/msup>+<\/mo>4<\/mn>a<\/mi>b<\/mi>x<\/mi>=<\/mo>–<\/mo>4<\/mn>a<\/mi>c<\/mi><\/math>\r\n
\r\n
\r\nAdding\u00a0 \r\nb<\/mi>2<\/mn><\/msup><\/math>\r\n \u00a0to both sides<\/p>\r\n

\r\n4<\/mn>a<\/mi>2<\/mn><\/msup>x<\/mi>2<\/mn><\/msup>+<\/mo>4<\/mn>a<\/mi>b<\/mi>x<\/mi>+<\/mo>b<\/mi>2<\/mn><\/msup>=<\/mo>b<\/mi>2<\/mn><\/msup>–<\/mo>4<\/mn>a<\/mi>c<\/mi><\/math>\r\n<\/p>\r\n

\r\n2<\/mn>a<\/mi>x<\/mi>+<\/mo>b<\/mi><\/mrow><\/mfenced>2<\/mn><\/msup>=<\/mo>b<\/mi>2<\/mn><\/msup>–<\/mo>4<\/mn>a<\/mi>c<\/mi><\/math>\r\n<\/p>\r\n

\r\n2<\/mn>a<\/mi>x<\/mi>+<\/mo>b<\/mi>=<\/mo>±<\/mo>b<\/mi>2<\/mn><\/msup>–<\/mo>4<\/mn>a<\/mi>c<\/mi><\/msqrt><\/math>\r\n<\/p>\r\n

\r\nx<\/mi>=<\/mo>–<\/mo>b<\/mi> <\/mo>±<\/mo> <\/mo>b<\/mi>2<\/mn><\/msup>–<\/mo>4<\/mn>a<\/mi>c<\/mi><\/msqrt><\/mrow>2<\/mn>a<\/mi><\/mrow><\/mfrac><\/math>\r\n<\/p>\r\n

Sum and Product of the roots:<\/strong><\/span><\/p>\r\n

If\u00a0 \r\nα<\/mi><\/math>\r\n \u00a0and\u00a0 \r\nβ<\/mi><\/math>\r\n \u00a0are the two roots of the above equation, then<\/p>\r\n

\r\nα<\/mi>=<\/mo>–<\/mo>b<\/mi>+<\/mo>b<\/mi>2<\/mn><\/msup>–<\/mo>4<\/mn>a<\/mi>c<\/mi><\/msqrt><\/mrow>2<\/mn>a<\/mi><\/mrow><\/mfrac>,<\/mo><\/math>\r\n \u00a0\u00a0 \r\nβ<\/mi>=<\/mo>–<\/mo>b<\/mi>±<\/mo>b<\/mi>2<\/mn><\/msup>–<\/mo>4<\/mn>a<\/mi>c<\/mi><\/msqrt><\/mrow>2<\/mn>a<\/mi><\/mrow><\/mfrac><\/math>\r\n<\/p>\r\n

Adding the above two,\u00a0<\/p>\r\n

Sum of the roots =\u00a0 \r\nα<\/mi>+<\/mo>β<\/mi>=<\/mo>–<\/mo>b<\/mi>+<\/mo>b<\/mi>2<\/mn><\/msup>–<\/mo>4<\/mn>a<\/mi>c<\/mi><\/msqrt> <\/mo>–<\/mo>b<\/mi>–<\/mo>b<\/mi>2<\/mn><\/msup>–<\/mo>4<\/mn>a<\/mi>c<\/mi><\/msqrt><\/mrow>2<\/mn>a<\/mi><\/mrow><\/mfrac><\/math>\r\n \r\n=<\/mo>–<\/mo>2<\/mn>b<\/mi><\/mrow>2<\/mn>a<\/mi><\/mrow><\/mfrac>=<\/mo>–<\/mo>b<\/mi>a<\/mi><\/mfrac><\/math>\r\n<\/p>\r\n

Product of the roots:<\/strong><\/span><\/p>\r\n

\r\nα<\/mi>β<\/mi>=<\/mo>–<\/mo>b<\/mi><\/mrow><\/mfenced>2<\/mn><\/msup>–<\/mo>b<\/mi>2<\/mn><\/msup>–<\/mo>4<\/mn>a<\/mi>c<\/mi><\/msqrt><\/mfenced>2<\/mn><\/msup><\/mrow>4<\/mn>a<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac><\/math>\r\n \u00a0 \r\n=<\/mo>b<\/mi>2<\/mn><\/msup>–<\/mo>b<\/mi>2<\/mn><\/msup>+<\/mo>4<\/mn>a<\/mi>c<\/mi><\/mrow>4<\/mn>a<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac>=<\/mo>4<\/mn>a<\/mi>c<\/mi><\/mrow>4<\/mn>a<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac>=<\/mo>c<\/mi>a<\/mi><\/mfrac><\/math>\r\n<\/p>\r\n

Nature of the roots:<\/strong><\/span><\/p>\r\n

\"\"<\/p>\r\n

Transformation of equations:<\/strong><\/span><\/p>\r\n

Let\u00a0 \r\nα<\/mi><\/math>\r\n and\u00a0 \r\nβ<\/mi><\/math>\r\n \u00a0are the roots of the equation\u00a0 \r\na<\/mi>x<\/mi>2<\/mn><\/msup>+<\/mo>b<\/mi>x<\/mi>+<\/mo>c<\/mi>=<\/mo>0<\/mn><\/math>\r\n<\/p>\r\n

To find the equation whose roots are :<\/p>\r\n

(i) Negative of the roots of the equation\u00a0 \r\na<\/mi>x<\/mi>2<\/mn><\/msup>+<\/mo>b<\/mi>x<\/mi>+<\/mo>c<\/mi>=<\/mo>0<\/mn><\/math>\r\n <\/strong><\/p>\r\n

The required roots are\u00a0 \r\n–<\/mo>α<\/mi><\/math>\r\n \u00a0 and\u00a0\u00a0 \r\n–<\/mo>β<\/mi><\/math>\r\n .<\/p>\r\n

This can be obtained by substituting<\/p>\r\n

\r\ny<\/mi>=<\/mo>–<\/mo>α<\/mi>=<\/mo>–<\/mo>x<\/mi> <\/mo><\/math>\r\n \u00a0 \u00a0 \r\n⇒<\/mo>x<\/mi>=<\/mo>–<\/mo>y<\/mi><\/math>\r\n<\/p>\r\n

so,\u00a0 \r\na<\/mi>–<\/mo>y<\/mi><\/mrow><\/mfenced>2<\/mn><\/msup>+<\/mo>b<\/mi>–<\/mo>y<\/mi><\/mrow><\/mfenced>+<\/mo>c<\/mi>=<\/mo>0<\/mn><\/math>\r\n \u00a0 \r\n⇒<\/mo>a<\/mi>y<\/mi>2<\/mn><\/msup>–<\/mo>b<\/mi>y<\/mi>+<\/mo>c<\/mi>=<\/mo>0<\/mn><\/math>\r\n<\/p>\r\n

or\u00a0 \u00a0 \u00a0\u00a0 \r\na<\/mi>x<\/mi>2<\/mn><\/msup>+<\/mo>b<\/mi>x<\/mi>+<\/mo>c<\/mi>=<\/mo>0<\/mn><\/math>\r\n<\/p>\r\n

(ii) Increased by\u00a0 \r\nh<\/mi><\/math>\r\n \u00a0i.e.\u00a0 \r\nα<\/mi>+<\/mo>h<\/mi> <\/mo>,<\/mo> <\/mo>β<\/mi>+<\/mo>h<\/mi><\/math>\r\n <\/strong><\/p>\r\n

substituting\u00a0 \r\ny<\/mi>=<\/mo>α<\/mi>+<\/mo>h<\/mi>=<\/mo>x<\/mi>+<\/mo>h<\/mi><\/math>\r\n \u00a0 \r\n⇒<\/mo>x<\/mi>=<\/mo>y<\/mi>–<\/mo>h<\/mi><\/math>\r\n<\/p>\r\n

so,\u00a0 \r\na<\/mi>y<\/mi>–<\/mo>h<\/mi><\/mrow><\/mfenced>2<\/mn><\/msup>+<\/mo>b<\/mi>y<\/mi>–<\/mo>h<\/mi><\/mrow><\/mfenced>+<\/mo>c<\/mi>=<\/mo>0<\/mn><\/math>\r\n<\/p>\r\n

\r\n⇒<\/mo>a<\/mi>y<\/mi>2<\/mn><\/msup>+<\/mo>y<\/mi>b<\/mi>–<\/mo>2<\/mn>a<\/mi>h<\/mi><\/mrow><\/mfenced>+<\/mo>a<\/mi>h<\/mi>2<\/mn><\/msup>–<\/mo>b<\/mi>h<\/mi>+<\/mo>c<\/mi><\/mrow><\/mfenced>=<\/mo>0<\/mn><\/math>\r\n<\/p>\r\n

so the required equation is\u00a0<\/p>\r\n

\r\na<\/mi>x<\/mi>2<\/mn><\/msup>+<\/mo>x<\/mi>b<\/mi>–<\/mo>2<\/mn>a<\/mi>h<\/mi><\/mrow><\/mfenced>+<\/mo>a<\/mi>h<\/mi>2<\/mn><\/msup>–<\/mo>b<\/mi>h<\/mi>+<\/mo>c<\/mi><\/mrow><\/mfenced>=<\/mo>0<\/mn><\/math>\r\n<\/p>\r\n

Sign of coefficient determining the sign of both real roots of\r\na<\/mi>x<\/mi>2<\/mn><\/msup>+<\/mo>b<\/mi>x<\/mi>+<\/mo>c<\/mi>=<\/mo>0<\/mn><\/math>\r\n<\/strong><\/span><\/p>\r\n

\"\"<\/p>","protected":false},"excerpt":{"rendered":"Roots of the equation\u00a0 ax2+bx+c=0 Multiplying the equation\u00a0 ax2+bx+c=0 \u00a0\u00a0both sides by\u00a0 4a , we get 4a2x2+4abx=–4ac Adding\u00a0 b2 \u00a0to both sides 4a2x2+4abx+b2=b2–4ac 2ax+b2=b2–4ac 2ax+b=±b2–4ac x=–b ± b2–4ac2a Sum and Product of the roots: If\u00a0 α \u00a0and\u00a0 β \u00a0are the two roots of the above equation, then α=–b+b2–4ac2a, \u00a0\u00a0 β=–b±b2–4ac2a Adding the above two,\u00a0 Sum of the […]","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-2129","knowledgebase","type-knowledgebase","status-publish","hentry","knowledgebase_cat-quadratics","no-wpautop"],"_links":{"self":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/2129","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase"}],"about":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/types\/knowledgebase"}],"author":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/comments?post=2129"}],"version-history":[{"count":16,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/2129\/revisions"}],"predecessor-version":[{"id":2150,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/2129\/revisions\/2150"}],"wp:attachment":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/media?parent=2129"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}